TrajPy: empowering feature engineering for trajectory analysis across domains.

Motivation: Trajectories, which are sequentially measured quantities that form a path, are an important presence in many different fields, from hadronic beams in physics to electrocardiograms in medicine. Trajectory analysis requires the quantification and classification of curves, either by using statistical descriptors or physics-based features. To date, no extensive and user-friendly package for trajectory analysis has been readily available, despite its importance and potential application across various domains. Results: We have developed TrajPy, a free, open-source Python package that serves as a complementary tool for empowering trajectory analysis. This package features a user-friendly graphical user interface and offers a set of physical descriptors that aid in characterizing these complex structures. TrajPy has already been successfully applied to studies of mitochondrial motility in neuroblastoma cell lines and the analysis of in silico models for cell migration, in combination with image analysis. Availability and implementation: The TrajPy package is developed in Python 3 and is released under the GNU GPL-3.0 license. It can easily be installed via PyPi, and the development source code is accessible at the repository: https://github.com/ocbe-uio/TrajPy/. The package release is also automatically archived with the DOI 10.5281/zenodo.3656044.

The ECM and tissue architecture are major determinants of early invasion mediated by E-cadherin dysfunction.

Germline mutations of E-cadherin cause Hereditary Diffuse Gastric Cancer (HDGC), a highly invasive cancer syndrome characterised by the occurrence of diffuse-type gastric carcinoma and lobular breast cancer. In this disease, E-cadherin-defective cells are detected invading the adjacent stroma since very early stages. Although E-cadherin loss is well established as a triggering event, other determinants of the invasive process persist largely unknown. Herein, we develop an experimental strategy that comprises in vitro extrusion assays using E-cadherin mutants associated to HDGC, as well as mathematical models epitomising epithelial dynamics and its interaction with the extracellular matrix (ECM). In vitro, we verify that E-cadherin dysfunctional cells detach from the epithelial monolayer and extrude basally into the ECM. Through phase-field modelling we demonstrate that, aside from loss of cell-cell adhesion, increased ECM attachment further raises basal extrusion efficiency. Importantly, by combining phase-field and vertex model simulations, we show that the cylindrical structure of gastric glands strongly promotes the cell's invasive ability. Moreover, we validate our findings using a dissipative particle dynamics simulation of epithelial extrusion. Overall, we provide the first evidence that cancer cell invasion is the outcome of defective cell-cell linkages, abnormal interplay with the ECM, and a favourable 3D tissue structure.

A mathematical model of fibrinogen-mediated erythrocyte-erythrocyte adhesion.

Erythrocytes are deformable cells that undergo progressive biophysical and biochemical changes affecting the normal blood flow. Fibrinogen, one of the most abundant plasma proteins, is a primary determinant for changes in haemorheological properties, and a major independent risk factor for cardiovascular diseases. In this study, the adhesion between human erythrocytes is measured by atomic force microscopy (AFM) and its effect observed by micropipette aspiration technique, in the absence and presence of fibrinogen. These experimental data are then used in the development of a mathematical model to examine the biomedical relevant interaction between two erythrocytes. Our designed mathematical model is able to explore the erythrocyte-erythrocyte adhesion forces and changes in erythrocyte morphology. AFM erythrocyte-erythrocyte adhesion data show that the work and detachment force necessary to overcome the adhesion between two erythrocytes increase in the presence of fibrinogen. The changes in erythrocyte morphology, the strong cell-cell adhesion and the slow separation of the two cells are successfully followed in the mathematical simulation. Erythrocyte-erythrocyte adhesion forces and energies are quantified and matched with experimental data. The changes observed on erythrocyte-erythrocyte interactions may give important insights about the pathophysiological relevance of fibrinogen and erythrocyte aggregation in hindering microcirculatory blood flow.

How abundant are superoxide and hydrogen peroxide in the vasculature lumen, how far can they reach?

Paracrine superoxide (O2•-) and hydrogen peroxide (H2O2) signaling critically depends on these substances' concentrations, half-lives and transport ranges in extracellular media. Here we estimated these parameters for the lumen of human capillaries, arterioles and arteries using reaction-diffusion-advection models. These models considered O2•- and H2O2 production by endothelial cells and uptake by erythrocytes and endothelial cells, O2•- dismutation, O2•- and H2O2 diffusion and advection by the blood flow. Results show that in this environment O2•- and H2O2 have half-lives <60. ms and <40. ms, respectively, the former determined by the plasma SOD3 activity, the latter by clearance by endothelial cells and erythrocytes. H2O2 concentrations do not exceed the 10 nM scale. Maximal O2•- concentrations near vessel walls exceed H2O2's several-fold when the latter results solely from O2•- dismutation. Cytosolic dismutation of inflowing O2•- may thus significantly contribute to H2O2 delivery to cells. O2•- concentrations near vessel walls decay to 50% of maximum 12 µm downstream from O2•- production sites. H2O2 concentrations in capillaries decay to 50% of maximum 22 µm (6.0 µm) downstream from O2•- (H2O2) production sites. Near arterioles' (arteries') walls, they decay by 50% within 6.0 µm (4. µm) of H2O2 production sites. However, they reach maximal values 50 µm (24 µm) downstream from O2•- production sites and decrease by 50% over 650 µm (500 µm). Arterial/olar endothelial cells might thus signal over a mm downstream through O2•--derived H2O2, though this requires nM-sensitive H2O2 transduction mechanisms.

A mathematical model for the dependence of keratin aggregate formation on the quantity of mutant keratin expressed in EGFP-K14 R125P keratinocytes.

We examined keratin aggregate formation and the possible mechanisms involved. With this aim, we observed the effect that different ratios between mutant and wild-type keratins expressed in cultured keratinocytes may have on aggregate formation in vitro, as well as how keratin aggregate formation affects the mechanical properties of cells at the cell cortex. To this end we prepared clones with expression rates as close as possible to 25%, 50% and 100% of the EGFP-K14 proteins (either WT or R125P and V270M mutants). Our results showed that only in the case of the 25% EGFP-K14 R125P mutant significant differences could be seen. Namely, we observed in this case the largest accumulation of keratin aggregates and a significant reduction in cell stiffness. To gain insight into the possible mechanisms behind this observation, we extended our previous mathematical model of keratin dynamics by implementing a more complex reaction network that considers the coexistence of wild-type and mutant keratins in the cell. The new model, consisting of a set of coupled, non-linear, ordinary differential equations, allowed us to draw conclusions regarding the relative amounts of intermediate filaments and aggregates in cells, and suggested that aggregate formation by asymmetric binding between wild-type and mutant keratins could explain the data obtained on cells grown in culture.

Intratumoral VEGF nanotrapper reduces gliobastoma vascularization and tumor cell mass.

Glioblastoma multiforme (GBM) is the most aggressive and invasive malignant brain cancer. GBM is characterized by a dramatic metabolic imbalance leading to increased secretion of the pro-angiogenic factor VEGF and subsequent abnormal tumor vascularization. In 2009, FDA approved the intravenous administration of bevacizumab, an anti-VEGF monoclonal antibody, as a therapeutic agent for patients with GBM. However, the number of systemic side effects and reduced accessibility of bevacizumab to the central nervous system and consequently to the GBM tumor mass limited its effectiveness in improving patient survival. In this study, we combined experimental and computational modelling to quantitatively characterize the dynamics of VEGF secretion and turnover in GBM and in normal brain cells and simultaneous monitoring of vessel growth. We showed that sequestration of VEGF inside GBM cells, can be used as a novel target for improved bevacizumab-based therapy. We have engineered the VEGF nanotrapper, a cargo system that allows cellular uptake of bevacizumab and inhibits VEGF secretion required for angiogenesis activation and development. Here, we show the therapeutic efficacy of this nanocargo in reducing vascularization and tumor cell mass of GBM in vitro and in vivo cancer models.

Cortical stiffness of keratinocytes measured by lateral indentation with optical tweezers.

Keratin intermediate filaments are the principal structural element of epithelial cells. Their importance in providing bulk cellular stiffness is well recognized, but their role in the mechanics of cell cortex is less understood. In this study, we therefore compared the cortical stiffness of three keratinocyte lines: primary wild type cells (NHEK2), immortalized wild type cells (NEB1) and immortalized mutant cells (KEB7). The cortical stiffness was measured by lateral indentation of cells with AOD-steered optical tweezers without employing any moving mechanical elements. The method was validated on fixed cells and Cytochalasin-D treated cells to ensure that the observed variations in stiffness within a single cell line were not a consequence of low measurement precision. The measurements of the cortical stiffness showed that primary wild type cells were significantly stiffer than immortalized wild type cells, which was also detected in previous studies of bulk elasticity. In addition, a small difference between the mutant and the wild type cells was detected, showing that mutation of keratin impacts also the cell cortex. Thus, our results indicate that the role of keratins in cortical stiffness is not negligible and call for further investigation of the mechanical interactions between keratins and elements of the cell cortex.

Adhesion modulates cell morphology and migration within dense fibrous networks.

One of the most fundamental abilities required for the sustainability of complex life forms is active cell migration, since it is essential in diverse processes from morphogenesis to leukocyte chemotaxis in immune response. The movement of a cell is the result of intricate mechanisms, that involve the coordination between mechanical forces, biochemical regulatory pathways and environmental cues. In particular, epithelial cancer cells have to employ mechanical strategies in order to migrate through the tissue's basement membrane and infiltrate the bloodstream during the invasion stage of metastasis. In this work we explore how mechanical interactions such as spatial restriction and adhesion affect migration of a self-propelled droplet in dense fibrous media. We have performed a systematic analysis using a phase-field model and we propose a novel approach to simulate cell migration with dissipative particle dynamics modelling. With this purpose we have measured in our simulation the cell's velocity and quantified its morphology as a function of the fibre density and of its adhesiveness to the matrix fibres. Furthermore, we have compared our results to a previous in vitro migration assay of fibrosarcoma cells in fibrous matrices. The results show good agreement between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions are able to capture the main features of the system. Our results indicate that adhesiveness is critical for cell migration, by modulating cell morphology in crowded environments and by enhancing cell velocity. In addition, our analysis suggests that matrix metalloproteinases (MMPs) play an important role as adhesiveness modulators. We propose that new assays should be carried out to address the role of adhesion and the effect of different MMPs in cell migration under confined conditions.

Notch signaling and taxis mechanisms regulate early stage angiogenesis: A mathematical and computational model.

During angiogenesis, new blood vessels sprout and grow from existing ones. This process plays a crucial role in organ development and repair, in wound healing and in numerous pathological processes such as cancer progression or diabetes. Here, we present a mathematical model of early stage angiogenesis that permits exploration of the relative importance of mechanical, chemical and cellular cues. Endothelial cells proliferate and move over an extracellular matrix by following external gradients of Vessel Endothelial Growth Factor, adhesion and stiffness, which are incorporated to a Cellular Potts model with a finite element description of elasticity. The dynamics of Notch signaling involving Delta-4 and Jagged-1 ligands determines tip cell selection and vessel branching. Through their production rates, competing Jagged-Notch and Delta-Notch dynamics determine the influence of lateral inhibition and lateral induction on the selection of cellular phenotypes, branching of blood vessels, anastomosis (fusion of blood vessels) and angiogenesis velocity. Anastomosis may be favored or impeded depending on the mechanical configuration of strain vectors in the ECM near tip cells. Numerical simulations demonstrate that increasing Jagged production results in pathological vasculatures with thinner and more abundant vessels, which can be compensated by augmenting the production of Delta ligands.

Resonances in the response of fluidic networks inherent to the cooperation between elasticity and bifurcations.

A global response function (GRF) of an elastic network is introduced as a generalization of the response function (RF) of a rigid network, relating the average flow along the network with the pressure difference at its extremes. The GRF can be used to explore the frequency behaviour of a fluid confined in a tree-like symmetric elastic network in which vessels bifurcate into identical vessels. We study such dynamic response for elastic vessel networks containing viscous fluids. We find that the bifurcation structure, inherent to tree-like networks, qualitatively changes the dynamic response of a single elastic vessel, and gives resonances at certain frequencies. This implies that the average flow throughout the network could be enhanced if the pulsatile forcing at the network's inlet were imposed at the resonant frequencies. The resonant behaviour comes from the cooperation between the bifurcation structure and the elasticity of the network, since the GRF has no resonances either for a single elastic vessel or for a rigid network. We have found that resonances shift to high frequencies as the system becomes more rigid. We have studied two different symmetric tree-like network morphologies and found that, while many features are independent of network morphology, particular details of the response are morphology dependent. Our results could have applications to some biophysical networks, for which the morphology could be approximated to a tree-like symmetric structure and a constant pressure at the outlet. The GRF for these networks is a characteristic of the system fluid-network, being independent of the dynamic flow (or pressure) at the network's inlet. It might therefore represent a good quantity to differentiate healthy vasculatures from those with a medical condition. Our results could also be experimentally relevant in the design of networks engraved in microdevices, since the limit of the rigid case is almost impossible to attain with the materials used in microfluidics and the condition of constant pressure at the outlet is often given by the atmospheric pressure.

Soft culture substrates favor stem-like cellular phenotype and facilitate reprogramming of human mesenchymal stem/stromal cells (hMSCs) through mechanotransduction.

Biophysical cues influence many aspects of cell behavior. Stiffness of the extracellular matrix is probed by cells and transduced into biochemical signals through mechanotransduction protein networks, strongly influencing stem cell behavior. Cellular stemness is intimately related with mechanical properties of the cell, like intracellular contractility and stiffness, which in turn are influenced by the microenvironment. Pluripotency is associated with soft and low-contractility cells. Hence, we postulated that soft cell culture substrates, presumably inducing low cellular contractility and stiffness, increase the reprogramming efficiency of mesenchymal stem/stromal cells (MSCs) into induced pluripotent stem cells (iPSCs). We demonstrate that soft substrates (1.5 or 15 kPa polydimethylsiloxane - PDMS) caused modulation of several cellular features of MSCs into a phenotype closer to pluripotent stem cells (PSCs). MSCs cultured on soft substrates presented more relaxed nuclei, lower maturation of focal adhesions and F-actin assembling, more euchromatic and less heterochromatic nuclear DNA regions, and increased expression of pluripotency-related genes. These changes correlate with the reprogramming of MSCs, with a positive impact on the kinetics, robustness of colony formation and reprogramming efficiency. Additionally, substrate stiffness influences several phenotypic features of iPS cells and colonies, and data indicates that soft substrates favor full iPSC reprogramming.

Localized redox relays as a privileged mode of cytoplasmic hydrogen peroxide signaling.

Hydrogen peroxide (H2O2) is a key signaling agent. Its best characterized signaling actions in mammalian cells involve the early oxidation of thiols in cytoplasmic phosphatases, kinases and transcription factors. However, these redox targets are orders of magnitude less H2O2-reactive and abundant than cytoplasmic peroxiredoxins. How can they be oxidized in a signaling time frame? Here we investigate this question using computational reaction-diffusion models of H2O2 signaling. The results show that at H2O2 supply rates commensurate with mitogenic signaling a H2O2 concentration gradient with a length scale of a few tenths of µm is established. Even near the supply sites H2O2 concentrations are far too low to oxidize typical targets in an early mitogenic signaling time frame. Furthermore, any inhibition of the peroxiredoxin or increase in H2O2 supply able to drastically increase the local H2O2 concentration would collapse the concentration gradient and/or cause an extensive oxidation of the peroxiredoxins I and II, inconsistent with experimental observations. In turn, the local concentrations of peroxiredoxin sulfenate and disulfide forms exceed those of H2O2 by several orders of magnitude. Redox targets reacting with these forms at rate constants much lower than that for, say, thioredoxin could be oxidized within seconds. Moreover, the spatial distribution of the concentrations of these peroxiredoxin forms allows them to reach targets within 1 µm from the H2O2 sites while maintaining signaling localized. The recruitment of peroxiredoxins to specific sites such as caveolae can dramatically increase the local concentrations of the sulfenic and disulfide forms, thus further helping these species to outcompete H2O2 for the oxidation of redox targets. Altogether, these results suggest that H2O2 signaling is mediated by localized redox relays whereby peroxiredoxins are oxidized to sulfenate and disulfide forms at H2O2 supply sites and these forms in turn oxidize the redox targets near these sites.

The Force at the Tip--Modelling Tension and Proliferation in Sprouting Angiogenesis.

Sprouting angiogenesis, where new blood vessels grow from pre-existing ones, is a complex process where biochemical and mechanical signals regulate endothelial cell proliferation and movement. Therefore, a mathematical description of sprouting angiogenesis has to take into consideration biological signals as well as relevant physical processes, in particular the mechanical interplay between adjacent endothelial cells and the extracellular microenvironment. In this work, we introduce the first phase-field continuous model of sprouting angiogenesis capable of predicting sprout morphology as a function of the elastic properties of the tissues and the traction forces exerted by the cells. The model is very compact, only consisting of three coupled partial differential equations, and has the clear advantage of a reduced number of parameters. This model allows us to describe sprout growth as a function of the cell-cell adhesion forces and the traction force exerted by the sprout tip cell. In the absence of proliferation, we observe that the sprout either achieves a maximum length or, when the traction and adhesion are very large, it breaks. Endothelial cell proliferation alters significantly sprout morphology, and we explore how different types of endothelial cell proliferation regulation are able to determine the shape of the growing sprout. The largest region in parameter space with well formed long and straight sprouts is obtained always when the proliferation is triggered by endothelial cell strain and its rate grows with angiogenic factor concentration. We conclude that in this scenario the tip cell has the role of creating a tension in the cells that follow its lead. On those first stalk cells, this tension produces strain and/or empty spaces, inevitably triggering cell proliferation. The new cells occupy the space behind the tip, the tension decreases, and the process restarts. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of forces in sprouting, hence underlining the necessary collaboration between modelling and molecular biology techniques to improve the current state-of-the-art.

Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply.

We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, which explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray's law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like in vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.

Hypoxia in vascular networks: a complex system approach to unravel the diabetic paradox.

In this work we model the extent of hypoxia in the diabetic retina as a function of the area affected by vessel disruption. We find two regimes that differ on the ratio between the area of disrupted vasculature and the area of tissue in hypoxia. In the first regime the hypoxia is localized in the vicinity of the vascular disruption, while in the second regime there is a generalized hypoxia in the affected tissue. The transition between these two regimes occurs when the tissue area affected by individual sites of vessel damage is on the order of the square of the characteristic irrigation length in the tissue (the maximum distance that an irrigated point in the tissue is from an existing vessel). We observe that very high levels of hypoxia are correlated with the rupture of larger vessels in the retina, and with smaller radii of individual sites of vessel damage. Based on this property of vascular networks, we propose a novel mechanism for the transition between the nonproliferative and the proliferative stages in diabetic retinopathy.

Flow and anastomosis in vascular networks.

We analyze the effect that the geometrical place of anastomosis in the circulatory tree has on blood flow. We introduce an idealized model that consists of a symmetric network for the arterial and venous vascular trees. We consider that the network contains a viscoelastic fluid with the rheological characteristics of blood, and analyze the network hydrodynamic response to a time-dependent periodic pressure gradient. This response is a measurement of the resistance to flow: the larger the response, the smaller the resistance to flow. We find that for networks whose vessels have the same radius and length, the outer the level of the branching tree in which anastomosis occurs, the larger the network response. Moreover, when anastomosis is incorporated in the form of bypasses that bridge vessels at different bifurcation levels, the further apart are the levels bridged by the bypass, the larger the response is. Furthermore, we apply the model to the available information for the dog circulatory system and find that the effect that anastomosis causes at different bifurcation levels is strongly determined by the structure of the underlying network without anastomosis. We rationalize our results by introducing two idealized models and approximated analytical expressions that allow us to argue that, to a large extent, the response of the network with anastomosis is determined locally. We have also considered the influence of the myogenic effect. This one has a large quantitative impact on the network response. However, the qualitative behavior of the network response with anastomosis is the same with or without consideration of the myogenic effect. That is, it depends on the structure that the underlying vessel network has in a small neighborhood around the place where anastomosis occurs. This implies that whenever there is an underlying tree-like network in an in vivo vasculature, our model is able to interpret the anastomotic effect.

Why do protein folding rates correlate with metrics of native topology?

For almost 15 years, the experimental correlation between protein folding rates and the contact order parameter has been under scrutiny. Here, we use a simple simulation model combined with a native-centric interaction potential to investigate the physical roots of this empirical observation. We simulate a large set of circular permutants, thus eliminating dependencies of the folding rate on other protein properties (e.g. stability). We show that the rate-contact order correlation is a consequence of the fact that, in high contact order structures, the contact order of the transition state ensemble closely mirrors the contact order of the native state. This happens because, in these structures, the native topology is represented in the transition state through the formation of a network of tertiary interactions that are distinctively long-ranged.