Mathematical modeling of eukaryotic cell migration: insights beyond experiments

Quantitative Motion Analysis in Two and Three Dimensions

This chapter describes 2D quantitative methods for motion analysis as well as 3D motion analysis and reconstruction methods. Emphasis is placed on the analysis of dynamic cell shape changes that occur through extension and retraction of force generating structures such as pseudopodia and lamellipodia. Quantitative analysis of these structures is an underutilized tool in the field of cell migration. Our intent, therefore, is to present methods that we developed in an effort to elucidate mechanisms of basic cell motility, directed cell motion during chemotaxis, and metastasis. We hope to demonstrate how application of these methods can more clearly define alterations in motility that arise due to specific mutations or disease and hence, suggest mechanisms or pathways involved in normal cell crawling and treatment strategies in the case of disease. In addition, we present a 4D tumorigenesis model for high-resolution analysis of cancer cells from cell lines and human cancer tissue in a 3D matrix. Use of this model led to the discovery of the coalescence of cancer cell aggregates and unique cell behaviors not seen in normal cells or normal tissue. Graphic illustrations to visually display and quantify cell shape are presented along with algorithms and formulae for calculating select 2D and 3D motion analysis parameters.

Quantifying Modes of 3D Cell Migration

Although it is widely appreciated that cells migrate in a variety of diverse environments in vivo, we are only now beginning to use experimental workflows that yield images with sufficient spatiotemporal resolution to study the molecular processes governing cell migration in 3D environments. Since cell migration is a dynamic process, it is usually studied via microscopy, but 3D movies of 3D processes are difficult to interpret by visual inspection. In this review, we discuss the technologies required to study the diversity of 3D cell migration modes with a focus on the visualization and computational analysis tools needed to study cell migration quantitatively at a level comparable to the analyses performed today on cells crawling on flat substrates.

Measuring fast stochastic displacements of bio-membranes with dynamic optical displacement spectroscopy

Stochastic displacements or fluctuations of biological membranes are increasingly recognized as an important aspect of many physiological processes, but hitherto their precise quantification in living cells was limited due to a lack of tools to accurately record them. Here we introduce a novel technique—dynamic optical displacement spectroscopy (DODS), to measure stochastic displacements of membranes with unprecedented combined spatiotemporal resolution of 20 nm and 10 μs. The technique was validated by measuring bending fluctuations of model membranes. DODS was then used to explore the fluctuations in human red blood cells, which showed an ATP-induced enhancement of non-Gaussian behaviour. Plasma membrane fluctuations of human macrophages were quantified to this accuracy for the first time. Stimulation with a cytokine enhanced non-Gaussian contributions to these fluctuations. Simplicity of implementation, and high accuracy make DODS a promising tool for comprehensive understanding of stochastic membrane processes.

Precise quantification of stochastic motions of biological membranes is limited by a lack of suitable detection methods. Here Monzel et al. develop dynamic optical displacement spectroscopy to measure stochastic membrane displacements at 20 nm/10 μs spatiotemporal resolution.

CPI motif interaction is necessary for capping protein function in cells

Capping protein (CP) has critical roles in actin assembly in vivo and in vitro. CP binds with high affinity to the barbed end of actin filaments, blocking the addition and loss of actin subunits. Heretofore, models for actin assembly in cells generally assumed that CP is constitutively active, diffusing freely to find and cap barbed ends. However, CP can be regulated by binding of the 'capping protein interaction' (CPI) motif, found in a diverse and otherwise unrelated set of proteins that decreases, but does not abolish, the actin-capping activity of CP and promotes uncapping in biochemical experiments. Here, we report that CP localization and the ability of CP to function in cells requires interaction with a CPI-motif-containing protein. Our discovery shows that cells target and/or modulate the capping activity of CP via CPI motif interactions in order for CP to localize and function in cells.

The jigsaw puzzle of sequence phenotype inference: Piecing together Shannon entropy, importance sampling, and Empirical Bayes

A nucleotide sequence 35 base pairs long can take 1,180,591,620,717,411,303,424 possible values. An example of systems biology datasets, protein binding microarrays, contain activity data from about 40,000 such sequences. The discrepancy between the number of possible configurations and the available activities is enormous. Thus, albeit that systems biology datasets are large in absolute terms, they oftentimes require methods developed for rare events due to the combinatorial increase in the number of possible configurations of biological systems. A plethora of techniques for handling large datasets, such as Empirical Bayes, or rare events, such as importance sampling, have been developed in the literature, but these cannot always be simultaneously utilized. Here we introduce a principled approach to Empirical Bayes based on importance sampling, information theory, and theoretical physics in the general context of sequence phenotype model induction. We present the analytical calculations that underlie our approach. We demonstrate the computational efficiency of the approach on concrete examples, and demonstrate its efficacy by applying the theory to publicly available protein binding microarray transcription factor datasets and to data on synthetic cAMP-regulated enhancer sequences. As further demonstrations, we find transcription factor binding motifs, predict the activity of new sequences and extract the locations of transcription factor binding sites. In summary, we present a novel method that is efficient (requiring minimal computational time and reasonable amounts of memory), has high predictive power that is comparable with that of models with hundreds of parameters, and has a limited number of optimized parameters, proportional to the sequence length.

The Arp2/3 inhibitory protein arpin induces cell turning by pausing cell migration

Branched actin networks generated by the Arp2/3 complex provide the driving force for leading edge protrusion in migrating cells. We recently identified Arpin, a protein that inhibits the Arp2/3 complex in lamellipodia. Arpin is activated by the small GTPase Rac, which triggers lamellipodium formation, and thus Arpin renders protrusions unstable. A conserved role of Arpin is to induce migrating cells to turn in different migration models. Here we investigated the mechanism by which Arpin controls directional persistence. For this analysis, we segmented migration trajectories into alternating phases of active migration and pauses, based on a speed threshold. Regardless of the threshold value, Arpin induced more frequent pausing, during which the cell was more likely to change the direction of its migration. Arpin simultaneously acts on cell speed and directional persistence, which are strongly coupled parameters. Induction of frequent pausing by Arpin is consistent with Arpin circuitry: by inhibiting the Arp2/3 complex as a response to Rac activation, Arpin antagonizes a positive feedback loop that sustains protrusions at the leading edge and maintains active migration. We propose the 'duration of active migration' as a useful proxy to measure feedbacks associated with cell migration.

Talin determines the nanoscale architecture of focal adhesions

Insight into how molecular machines perform their biological functions depends on knowledge of the spatial organization of the components, their connectivity, geometry, and organizational hierarchy. However, these parameters are difficult to determine in multicomponent assemblies such as integrin-based focal adhesions (FAs). We have previously applied 3D superresolution fluorescence microscopy to probe the spatial organization of major FA components, observing a nanoscale stratification of proteins between integrins and the actin cytoskeleton. Here we combine superresolution imaging techniques with a protein engineering approach to investigate how such nanoscale architecture arises. We demonstrate that talin plays a key structural role in regulating the nanoscale architecture of FAs, akin to a molecular ruler. Talin diagonally spans the FA core, with its N terminus at the membrane and C terminus demarcating the FA/stress fiber interface. In contrast, vinculin is found to be dispensable for specification of FA nanoscale architecture. Recombinant analogs of talin with modified lengths recapitulated its polarized orientation but altered the FA/stress fiber interface in a linear manner, consistent with its modular structure, and implicating the integrin-talin-actin complex as the primary mechanical linkage in FAs. Talin was found to be ∼97 nm in length and oriented at ∼15° relative to the plasma membrane. Our results identify talin as the primary determinant of FA nanoscale organization and suggest how multiple cellular forces may be integrated at adhesion sites.

Model for adhesion clutch explains biphasic relationship between actin flow and traction at the cell leading edge

Cell motility relies on the continuous reorganization of a dynamic actin-myosin-adhesion network at the leading edge of the cell, in order to generate protrusion at the leading edge and traction between the cell and its external environment. We analyze experimentally measured spatial distributions of actin flow, traction force, myosin density, and adhesion density in control and pharmacologically perturbed epithelial cells in order to develop a mechanical model of the actin-adhesion-myosin self-organization at the leading edge. A model in which the F-actin network is treated as a viscous gel, and adhesion clutch engagement is strengthened by myosin but weakened by actin flow, can explain the measured molecular distributions and correctly predict the spatial distributions of the actin flow and traction stress. We test the model by comparing its predictions with measurements of the actin flow and traction stress in cells with fast and slow actin polymerization rates. The model predicts how the location of the lamellipodium-lamellum boundary depends on the actin viscosity and adhesion strength. The model further predicts that the location of the lamellipodium-lamellum boundary is not very sensitive to the level of myosin contraction.

Cost-benefit analysis of the mechanisms that enable migrating cells to sustain motility upon changes in matrix environments

Cells can move through extracellular environments with varying geometries and adhesive properties. Adaptation to these differences is achieved by switching between different modes of motility, including lamellipod-driven and blebbing motility. Further, cells can modulate their level of adhesion to the extracellular matrix (ECM) depending on both the level of force applied to the adhesions and cell intrinsic biochemical properties. We have constructed a computational model of cell motility to investigate how motile cells transition between extracellular environments with varying surface continuity, confinement and adhesion. Changes in migration strategy are an emergent property of cells as the ECM geometry and adhesion changes. The transition into confined environments with discontinuous ECM fibres is sufficient to induce shifts from lamellipod-based to blebbing motility, while changes in confinement alone within a continuous geometry are not. The geometry of the ECM facilitates plasticity, by inducing shifts where the cell has high marginal gain from a mode change, and conserving persistency where the cell can continue movement regardless of the motility mode. This regulation of cell motility is independent of global changes in cytoskeletal properties, but requires locally higher linkage between the actin network and the plasma membrane at the cell rear, and changes in internal cell pressure. In addition to matrix geometry, we consider how cells might transition between ECM of different adhesiveness. We find that this requires positive feedback between the forces cells apply on the adhesion points, and the strength of the cell-ECM adhesions on those sites. This positive feedback leads to the emergence of a small number of highly adhesive cores, similar to focal adhesions. While the range of ECM adhesion levels the cell can invade is expanded with this feedback mechanism; the velocities are lowered for conditions where the positive feedback is not vital. Thus, plasticity of cell motility sacrifices the benefits of specialization, for robustness.

A mathematical model of collective cell migration in a three-dimensional, heterogeneous environment

Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed in vivo, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of Drosophila oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.

Cells on film – the past and future of cinemicroscopy

Movie making is now a ubiquitous experimental tool that biologists use alongside more traditional techniques such as molecular biology and biochemistry. It is no longer just cell biologists, but scientists from many other disciplines, such as immunology and neuroscience, that utilise movies to dissect their processes of interest. When did filming become such a standard laboratory technique? Who developed the use of the movie as an experimental tool? The Wellcome Library has recently restored and digitized a number of original 16-mm films from two pioneering cinemicroscopists, Ronald Canti and Michael Abercrombie, which are now freely available to the scientific community. In light of these films, this Essay will give a brief history of the early cinemicroscopists and discuss what is driving the use of movies in the laboratory today.

Microtubules stabilize cell polarity by localizing rear signals

Microtubules are known to play an important role in cell polarity; however, the mechanism remains unclear. Using cells migrating persistently on micropatterned strips, we found that depolymerization of microtubules caused cells to change from persistent to oscillatory migration. Mathematical modeling in the context of a local-excitation-global-inhibition control mechanism indicated that this mechanism can account for microtubule-dependent oscillation, assuming that microtubules remove inhibitory signals from the front after a delayed generation. Experiments further supported model predictions that the period of oscillation positively correlates with cell length and that oscillation may be induced by inhibiting retrograde motors. We suggest that microtubules are required not for the generation but for the maintenance of cell polarity, by mediating the global distribution of inhibitory signals. Disassembly of microtubules induces cell oscillation by allowing inhibitory signals to accumulate at the front, which stops frontal protrusion and allows the polarity to reverse.

Steering cell migration: lamellipodium dynamics and the regulation of directional persistence

Membrane protrusions at the leading edge of cells, known as lamellipodia, drive cell migration in many normal and pathological situations. Lamellipodial protrusion is powered by actin polymerization, which is mediated by the actin-related protein 2/3 (ARP2/3)-induced nucleation of branched actin networks and the elongation of actin filaments. Recently, advances have been made in our understanding of positive and negative ARP2/3 regulators (such as the SCAR/WAVE (SCAR/WASP family verprolin-homologous protein) complex and Arpin, respectively) and of proteins that control actin branch stability (such as glial maturation factor (GMF)) or actin filament elongation (such as ENA/VASP proteins) in lamellipodium dynamics and cell migration. This Review highlights how the balance between actin filament branching and elongation, and between the positive and negative feedback loops that regulate these activities, determines lamellipodial persistence. Importantly, directional persistence, which results from lamellipodial persistence, emerges as a critical factor in steering cell migration.