Measuring population-level plant gene flow with topological data analysis

Hybridization is an important factor influencing the evolution of many plant species. Measurement of interspecific gene flow and quantification of recombination provide insight into the processes that shape population genetic diversity. Recently, a new approach called topological data analysis (TDA) has begun to be implemented to track genomic recombination between species and populations. However, existing TDA methods tend to underestimate gene flow in cross-population models. Here, we present a novel graph-based approach to measuring gene flow, which occurs during ongoing speciation and hybridization in plants in various mating systems. We combine minimal spanning networks with topological filtering and then test the resulting parameter measurements on simulated primary divergence and secondary contact processes under different mating conditions. The resulting parameters, based on the first Betti number (total number of one-dimensional cycles), showed a high positive correlation with the number of migrants (N_m) in almost all cases of secondary contact. Considering the modularity of MSN networks, we have devised a parameter that works well in the event of primary divergence, where gene flow is at its highest level. This approach is suitable for various types of data, including SNPs, microsatellites, and binary restriction fragment data.     

A hybrid bioregulatory model of angiogenesis during bone fracture healing

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.     

A Continuum Mathematical Model of the Developing Murine Retinal Vasculature

Angiogenesis, the process of new vessel growth from pre-existing vasculature, is crucial in many biological situations such as wound healing and embryogenesis. Angiogenesis is also a key regulator of pathogenesis in many clinically important disease processes, for instance, solid tumour progression and ocular diseases. Over the past 10–20 years, tumour-induced angiogenesis has received a lot of attention in the mathematical modelling community and there have also been some attempts to model angiogenesis during wound healing. However, there has been little modelling work of vascular growth during normal development. In this paper, we describe an in silico representation of the developing retinal vasculature in the mouse, using continuum mathematical models consisting of systems of partial differential equations. The equations describe the migratory response of cells to growth factor gradients, the evolution of the capillary blood vessel density, and of the growth factor concentration. Our approach is closely coupled to an associated experimental programme to parameterise our model effectively and the simulations provide an excellent correlation with in vivo experimental data. Future work and development of this model will enable us to elucidate the impact of molecular cues upon vasculature development and the implications for eye diseases such as diabetic retinopathy and neonatal retinopathy of prematurity.     

Numerical modelling of the angiogenesis process in wound contraction

Angiogenesis consists of the growth of new blood vessels from the pre-existing vasculature. This phenomenon takes place in several biological processes, including wound healing. In this work, we present a mathematical model of angiogenesis applied to skin wound healing. The developed model includes biological (capillaries and fibroblasts), chemical (oxygen and angiogenic growth factor concentrations) and mechanical factors (cell traction forces and extracellular matrix deformation) that influence the evolution of the healing process. A novelty from previous works, apart from the coupling of angiogenesis and wound contraction, is the more realistic modelling of skin as a hyperelastic material. Large deformations are addressed using an updated Lagrangian approach. The coupled non-linear model is solved with the finite element method, and the process is studied over two wound geometries (circular and elliptical) of the same area. The results indicate that the elliptical wound vascularizes two days earlier than the circular wound but that they experience a similar contraction level, reducing its size by 25 %.     

Role of eNOS in neovascularization: NO for endothelial progenitor cells

Nitric oxide (NO) is a gaseous molecule with an astonishingly wide range of physiological and pathophysiological activities, including the regulation of vessel tone and angiogenesis in wound healing, inflammation, ischaemic cardiovascular diseases and malignant diseases. Recent data have revealed the predominant role of endothelial nitric oxide synthase (eNOS), an endothelial-cell-specific isoform of NO producing enzyme, in both angiogenesis (the development of new blood vessels derived from existing vessels) and vasculogenesis (blood vessel formation de novo from progenitor cells). In addition, successes in gene therapy, together with the recent development of an eNOS-specific inhibitor, suggest that the modulation of eNOS might be a potent new strategy for the control of pathological neovascularization.     

The angiogenesis inhibitor endostatin impairs blood vessel maturation during wound healing.

Endostatin is a cleavage product of collagen XVIII that strongly inhibits tumor angiogenesis. To determine if endostatin affects other angiogenic processes, we generated full-thickness excisional wounds on the back of mice that were systemically treated with recombinant murine endostatin. No macroscopic abnormalities of the wound healing process were observed. Histological analysis revealed normal wound contraction and re-epithelialization, but a slight reduction in granulation tissue formation and reduced matrix deposition at the wound edge. The blood vessel density in the wounds of endostatin-treated mice was not affected. However, ultrastructural analysis demonstrated severe abnormalities in blood vessel maturation. The wound vessels in the endostatin-treated mice were narrowed or closed with an irregular luminal surface, resulting in a severe reduction in the number of functional vessels and extravasation of erythrocytes. Endostatin treatment did not affect the expression level and localization of collagen XVIII mRNA and protein. Furthermore, the angiogenesis regulators vascular endothelial growth factor, angiopoietin-1, and angiopoietin-2 were normally expressed in the wounds of endostatin-treated mice. However, expression of the major wound matrix proteins fibronectin and collagens I and III was significantly reduced. This reduction is likely to explain the reduced density of the wound matrix. Our results demonstrate that endostatin treatment reduces the number of functional blood vessels and the matrix density in the granulation tissue, but does not significantly affect the overall wound healing process.     

Physical Model of the Dynamic Instability in an Expanding Cell Culture

Collective cell migration is of great significance in many biological processes. The goal of this work is to give a physical model for the dynamics of cell migration during the wound healing response. Experiments demonstrate that an initially uniform cell-culture monolayer expands in a nonuniform manner, developing fingerlike shapes. These fingerlike shapes of the cell culture front are composed of columns of cells that move collectively. We propose a physical model to explain this phenomenon, based on the notion of dynamic instability. In this model, we treat the first layers of cells at the front of the moving cell culture as a continuous one-dimensional membrane (contour), with the usual elasticity of a membrane: curvature and surface-tension. This membrane is active, due to the forces of cellular motility of the cells, and we propose that this motility is related to the local curvature of the culture interface; larger convex curvature correlates with a stronger cellular motility force. This shape-force relation gives rise to a dynamic instability, which we then compare to the patterns observed in the wound healing experiments.     

When the optimal is not the best: parameter estimation in complex biological models

Background

The vast computational resources that became available during the past decade enabled the development and simulation of increasingly complex mathematical models of cancer growth. These models typically involve many free parameters whose determination is a substantial obstacle to model development. Direct measurement of biochemical parameters in vivo is often difficult and sometimes impracticable, while fitting them under data-poor conditions may result in biologically implausible values.

Results

We discuss different methodological approaches to estimate parameters in complex biological models. We make use of the high computational power of the Blue Gene technology to perform an extensive study of the parameter space in a model of avascular tumor growth. We explicitly show that the landscape of the cost function used to optimize the model to the data has a very rugged surface in parameter space. This cost function has many local minima with unrealistic solutions, including the global minimum corresponding to the best fit.

Conclusions

The case studied in this paper shows one example in which model parameters that optimally fit the data are not necessarily the best ones from a biological point of view. To avoid force-fitting a model to a dataset, we propose that the best model parameters should be found by choosing, among suboptimal parameters, those that match criteria other than the ones used to fit the model. We also conclude that the model, data and optimization approach form a new complex system and point to the need of a theory that addresses this problem more generally.     

Regulation of angiogenesis: wound healing as a model

Normal tissue function requires adequate supply of oxygen through blood vessels. Understanding how blood vessels form is a challenging objective because angiogenesis is vital to many physiological and pathological processes. Unraveling mechanisms of angiogenesis would offer therapeutic options to ameliorate disorders that are currently leading causes of mortality and morbidity, including cardiovascular diseases, cancer, chronic inflammatory disorders, diabetic retinopathy, excessive tissue defects, and chronic non-healing wounds. Restoring blood flow to the site of injured tissue is a prerequisite for mounting a successful repair response, and wound angiogenesis represents a paradigmatic model to study molecular mechanisms involved in the formation and remodeling of vascular structures. In particular, repair of skin defects offers an ideal model to analyze angiogenesis due to its easy accessibility to control and manipulate this process. Most of those growth factors, extracellular matrix molecules, and cell types, recently discovered and considered as crucial factors in blood vessel formation, have been identified and analyzed during skin repair and the process of wound angiogenesis. This article will review cellular and molecular mechanisms controlling angiogenesis in cutaneous tissue repair in light of recent reports and data from our laboratories. In this article we will discuss the contribution of growth factors, basement membrane molecules, and mural cells in wound angiogenesis. The article provides a rationale for targeting the angiogenic response in order to modulate the outcome of the healing response.     

reg6 is required for branching morphogenesis during blood vessel regeneration in zebrafish caudal fins

Postnatal neovascularization is essential for wound healing, cancer progression, and many other physiological functions. However, its genetic mechanism is largely unknown. In this report, we study neovascularization in regenerating adult zebrafish fins using transgenic fish that express EGFP in blood vessel endothelial cells. We first describe the morphogenesis of regenerating vessels in wild-type animals and then the phenotypic analysis of a genetic mutation that disrupts blood vessel regeneration. In wild-type zebrafish caudal fins, amputated blood vessels heal their ends by 24 h postamputation (hpa) and then reconnect arteries and veins via anastomosis, to resume blood flow at wound sites by 48 hpa. The truncated vessels regenerate by first growing excess vessels to form unstructured plexuses, resembling the primary capillary plexuses formed during embryonic vasculogenesis. Interestingly, this mode of vessel growth switches by 8 days postamputation (dpa) to growth without a plexus intermediate. During blood vessel regeneration, vessel remodeling begins during early plexus formation and continues until the original vasculature pattern is reestablished at approximately 35 dpa. Temperature-sensitive mutants for reg6 have profound defects in blood vessel regeneration. At the restrictive temperature, reg6 regenerating blood vessels first fail to make reconnections between severed arteries and veins, and then form enlarged vascular sinuses rather than branched vascular plexuses. Reciprocal temperature-shift experiments show that reg6 function is required throughout plexus formation, but not during later growth. Our results suggest that the reg6 mutation causes defects in branch formation and/or angiogenic sprouting.     

Nestin‐expressing interfollicular blood vessel network contributes to skin transplant survival and wound healing

Using nestin‐driven green fluorescent protein (ND‐GFP) transgenic mice, we previously demonstrated an inter‐hair‐follicle blood vessel network that expresses ND‐GFP and appears to originate from ND‐GFP expressing hair‐follicle stem cells. We report here that angiogenesis of transplanted skin or healing wounds originates from this ND‐GFP‐expressing microvasculature network. ND‐GFP‐expressing blood vessels were visualized growing from the ND‐GFP‐expressing hair‐follicle stem cell area and re‐establishing the dermal microvasculature network after skin transplantation or wound healing. When the ND‐GFP stem cell area from the vibrissa (whisker) from ND‐GFP mice was transplanted to transgenic mice ubiquitously expressing RFP, we observed chimeric ND‐GFP‐RFP blood vessels, suggesting the joining of inter‐follicular blood vessel networks from the transplant and host. These observations suggest that the inter‐hair‐follicle blood‐vessel network contributes to skin transplant survival and wound healing. J. Cell. Biochem. 110: 80–86, 2010. © 2010 Wiley‐Liss, Inc.     

Galvanic microparticles increase migration of human dermal fibroblasts in a wound-healing model via reactive oxygen species pathway

Electrical signals have been implied in many biological mechanisms, including wound healing, which has been associated with transient electrical currents not present in intact skin. One method to generate electrical signals similar to those naturally occurring in wounds is by supplementation of galvanic particles dispersed in a cream or gel. We constructed a three-layered model of skin consisting of human dermal fibroblasts in hydrogel (mimic of dermis), a hydrogel barrier layer (mimic of epidermis) and galvanic microparticles in hydrogel (mimic of a cream containing galvanic particles applied to skin). Using this model, we investigated the effects of the properties and amounts of Cu/Zn galvanic particles on adult human dermal fibroblasts in terms of the speed of wound closing and gene expression. The collected data suggest that the effects on wound closing are due to the ROS-mediated enhancement of fibroblast migration, which is in turn mediated by the BMP/SMAD signaling pathway. These results imply that topical low-grade electric currents via microparticles could enhance wound healing.     

Identification of MrgX2 as a human G-protein-coupled receptor for proadrenomedullin N-terminal peptides

Pleiotrophin (PTN) is a heparin-binding growth/differentiation inducing cytokine that shares 50% amino acid sequence identity and striking domain homology with Midkine (MK), the only other member of the Ptn/Mk developmental gene family. The Ptn gene is expressed in sites of early vascular development in embryos and in healing wounds and its constitutive expression in many human tumors is associated with an angiogenic phenotype, suggesting that PTN has an important role in angiogenesis during development and in wound repair and advanced malignancies. To directly test whether PTN is angiogenic in vivo, we injected a plasmid to express PTN into ischemic myocardium in rats. Pleiotrophin stimulated statistically significant increases in both normal appearing new capillaries and arterioles each of which had readily detectable levels of the arteriole marker, smooth muscle cell alpha-actin. Furthermore, the newly formed blood vessels were shown to interconnect with the existent coronary vascular system. The results of these studies demonstrate directly that PTN is an effective angiogenic agent in vivo able to initiate new vessel formation that is both normal in appearance and function. The data suggest that PTN signals the more "complete" new blood vessel formation through its ability to stimulate different functions in different cell types not limited to the endothelial cell.     

Inferences from FRAP data are model dependent: A subdiffusive analysis

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.     

Retracted: Paeoniflorin promotes angiogenesis and tissue regeneration in a full-thickness cutaneous wound model through the PI3K/AKT pathway

The treatment of wounds remains a clinical challenge because of poor angiogenesis under the wound bed, and increasingly, the patients' need for functional and aesthetically pleasing scars. For the wound healing process, new blood vessels which can deliver nutrients and oxygen to the wound area are necessary. In this study, we investigated the pro-angiogenesis ability and mechanism in wound healing of paeoniflorin (PF), which is a traditional Chinese medicine. In our in vitro results, the ability for proliferation, migration and in vitro angiogenesis in human umbilical vein endothelial cells was promoted by coculturing with PF (1.25–5 μM). Meanwhile, molecular docking studies revealed that PF has excellent binding abilities to phosphatidylinositol-3-kinase (PI3K) and protein kinase B (AKT), and consistent with our western blot results, that PF suppressed PI3K and AKT phosphorylation. Furthermore, to investigate the healing effect of PF in vivo, we constructed a full-thickness cutaneous wound model in rats. PF stimulated the cellular proliferation status, collagen matrix deposition and remodeling processes in vitro and new blood vessel formation at the wound bed resulting in efficient wound healing after intragastric administration of 10 mg·kg⁻¹·day⁻¹ in vivo. Overall, PF performed the pro-angiogenetic effect in vitro and accelerating wound healing in vivo. In summary, the capacity for angiogenesis in endothelial cells could be enhanced by PF treatment via the PI3K/AKT pathway in vitro and could accelerate the wound healing process in vivo through collagen deposition and angiogenesis in regenerated tissue. This study provides evidence that application of PF represents a novel therapeutic approach for the treatment of cutaneous wounds.     

Agent-based modelling and parameter sensitivity analysis with a finite-element method for skin contraction

In this paper, we extend the model of wound healing by Boon et al. (J Biomech 49(8):1388–1401, 2016). In addition to explaining the model explicitly regarding every component, namely cells, signalling molecules and tissue bundles, we categorized fibroblasts as regular fibroblasts and myofibroblasts. We do so since it is widely documented that myofibroblasts play a significant role during wound healing and skin contraction and that they are the main phenotype of cells that is responsible for the permanent deformations. Furthermore, we carried out some sensitivity tests of the model by modifying certain parameter values, and we observe that the model shows some consistency with several biological phenomena. Using Monte Carlo simulations, we found that there is a significant strong positive correlation between the final wound area and the minimal wound area. The high correlation between the wound area after 4 days and the final/minimal wound area makes it possible for physicians to predict the most probable time evolution of the wound of the patient. However, the collagen density ratio at the time when the wound area reaches its equilibrium and minimum, cannot indicate the degree of wound contractions, whereas at the 4th day post-wounding, when the collagen is accumulating from null, there is a strong negative correlation between the area and the collagen density ratio. Further, under the circumstances that we modelled, the probability that patients will end up with 5% contraction is about 0.627.

     

Application of 1D blood flow models of the human arterial network to differential pressure predictions

A new application of 1D models of the human arterial network is proposed. We take advantage of the sensitivity of the models predictions for the pressure profiles within the main aorta to key model parameter values. We propose to use the patterns in the predicted differences from a base case as a way to infer to the most probable changes in the parameter values. We demonstrate this application using an impedance model that we have recently developed (Johnson, 2010). The input model parameters are all physiologically related, such as the geometric dimensions of large arteries, various blood properties, vessel elasticity, etc. and can therefore be patient specific. As a base case, nominal values from the literature are used. The necessary information to characterize the smaller arteries, arterioles, and capillaries is taken from a physical scaling model (West, 1999). Model predictions for the effective impedance of the human arterial system closely agree with experimental data available in the literature. The predictions for the pressure wave development along the main arteries are also found in qualitative agreement with previous published results. The model has been further validated against our own measured pressure data in the carotid and radial arteries, obtained from healthy individuals. Upon changes in the value of key model parameters, we show that the differences seen in the pressure profiles correspond to qualitatively different patterns for different parameters. This suggests the possibility of using the model in interpreting multiple pressure data of healthy/diseased individuals.     

Tracking collective cell motion by topological data analysis

By modifying and calibrating an active vertex model to experiments, we have simulated numerically a confluent cellular monolayer spreading on an empty space and the collision of two monolayers of different cells in an antagonistic migration assay. Cells are subject to inertial forces and to active forces that try to align their velocities with those of neighboring ones. In agreement with experiments in the literature, the spreading test exhibits formation of fingers in the moving interfaces, there appear swirls in the velocity field, and the polar order parameter and the correlation and swirl lengths increase with time. Numerical simulations show that cells inside the tissue have smaller area than those at the interface, which has been observed in recent experiments. In the antagonistic migration assay, a population of fluidlike Ras cells invades a population of wild type solidlike cells having shape parameters above and below the geometric critical value, respectively. Cell mixing or segregation depends on the junction tensions between different cells. We reproduce the experimentally observed antagonistic migration assays by assuming that a fraction of cells favor mixing, the others segregation, and that these cells are randomly distributed in space. To characterize and compare the structure of interfaces between cell types or of interfaces of spreading cellular monolayers in an automatic manner, we apply topological data analysis to experimental data and to results of our numerical simulations. We use time series of data generated by numerical simulations to automatically group, track and classify the advancing interfaces of cellular aggregates by means of bottleneck or Wasserstein distances of persistent homologies. These techniques of topological data analysis are scalable and could be used in studies involving large amounts of data. Besides applications to wound healing and metastatic cancer, these studies are relevant for tissue engineering, biological effects of materials, tissue and organ regeneration.     

A multiphase model describing vascular tumour growth

In this paper we present a new model framework for studying vascular tumour growth, in which the blood vessel density is explicitly considered. Our continuum model comprises conservation of mass and momentum equations for the volume fractions of tumour cells, extracellular material and blood vessels. We include the physical mechanisms that we believe to be dominant, namely birth and death of tumour cells, supply and removal of extracellular fluid via the blood and lymph drainage vessels, angiogenesis and blood vessel occlusion. We suppose that the tumour cells move in order to relieve the increase in mechanical stress caused by their proliferation. We show how to reduce the model to a system of coupled partial differential equations for the volume fraction of tumour cells and blood vessels and the phase averaged velocity of the mixture. We consider possible parameter regimes of the resulting model. We solve the equations numerically in these cases, and discuss the resulting behaviour. The model is able to reproduce tumour structure that is found in vivo in certain cases. Our framework can be easily modified to incorporate the effect of other phases, or to include the effect of drugs.     

Live imaging of wound angiogenesis reveals macrophage orchestrated vessel sprouting and regression

Wound angiogenesis is an integral part of tissue repair and is impaired in many pathologies of healing. Here, we investigate the cellular interactions between innate immune cells and endothelial cells at wounds that drive neoangiogenic sprouting in real time and in vivo. Our studies in mouse and zebrafish wounds indicate that macrophages are drawn to wound blood vessels soon after injury and are intimately associated throughout the repair process and that macrophage ablation results in impaired neoangiogenesis. Macrophages also positively influence wound angiogenesis by driving resolution of anti‐angiogenic wound neutrophils. Experimental manipulation of the wound environment to specifically alter macrophage activation state dramatically influences subsequent blood vessel sprouting, with premature dampening of tumour necrosis factor‐α expression leading to impaired neoangiogenesis. Complementary human tissue culture studies indicate that inflammatory macrophages associate with endothelial cells and are sufficient to drive vessel sprouting via vascular endothelial growth factor signalling. Subsequently, macrophages also play a role in blood vessel regression during the resolution phase of wound repair, and their absence, or shifted activation state, impairs appropriate vessel clearance.