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Alterations in mitochondrial dynamics, including their trafficking, can present early manifestation of neuronal degeneration. However, current methodologies used to study mitochondrial trafficking events rely on parameters that are mostly altered in later stages of neurodegeneration. Our objective was to establish a reliable computational methodology to detect early alterations in neuronal mitochondrial trafficking. We propose a novel quantitative analysis of mitochondria trajectories based on innovative movement descriptors, including straightness, efficiency, anisotropy, and kurtosis. Using biological data from differentiated SH-SY5Y cells treated with mitochondrial toxicants 6-hydroxydopamine and rotenone, we evaluated time and dose-dependent alterations in trajectory descriptors. Mitochondrial movement was analyzed by total internal reflection fluorescence microscopy followed by computer modelling to describe the process. The stacks of individual images were analyzed by an open source MATLAB algorithm (www.github.com/kandelj/MitoSPT) and to characterize mitochondria trajectories, we used the Python package trajpy (https://github.com/ocbe-uio/trajpy/). Our results confirm that this computational approach is effective and accurate in order to study mitochondrial motility and trajectories in the context of healthy and diseased neurons in different stages.

The dynamics of tumor growth and associated events cover multiple time and spatial scales, generally including extracellular, cellular and intracellular modifications. The main goal of this study is to model the biological and physical behavior of tumor evolution in presence of normal healthy tissue, considering a variety of events involved in the process. These include hyper and hypoactivation of signaling pathways during tumor growth, vessels’ growth, intratumoral vascularization and competition of cancer cells with healthy host tissue. The work addresses two distinctive phases in tumor development—the avascular and vascular phases—and in each stage two cases are considered—with and without normal healthy cells. The tumor growth rate increases considerably as closed vessel loops (anastomoses) form around the tumor cells resulting from tumor induced vascularization. When taking into account the host tissue around the tumor, the results show that competition between normal cells and cancer cells leads to the formation of a hypoxic tumor core within a relatively short period of time. Moreover, a dense intratumoral vascular network is formed throughout the entire lesion as a sign of a high malignancy grade, which is consistent with reported experimental data for several types of solid carcinomas. In comparison with other mathematical models of tumor development, in this work we introduce a multiscale simulation that models the cellular interactions and cell behavior as a consequence of the activation of oncogenes and deactivation of gene signaling pathways within each cell. Simulating a therapy that blocks relevant signaling pathways results in the prevention of further tumor growth and leads to an expressive decrease in its size (82% in the simulation)., Author summary Mathematical modeling and simulation of cancer across different biological scales is becoming increasingly important in the development of therapeutic strategies. In the current work, a multiscale model is presented to study the growth and progression of tumor and angiogenesis based on tumor-host interactions which allows investigating the effects of tumor-targeted therapy. Considering the signal-transduction networks involved in various types of cancers, we proposed a cascade that encompasses some significant signaling pathways. A Boolean network model is employed to describe the receptors cross-talk. As a result of the activation of oncogenes and deactivation of pertinent gene signaling pathways within each cell, the cellular interactions and cell behavior are modeled. By linking cells state with environmental cues, the tumor morphology is determined. Consistent with the experimental observations, the intratumoral vascularization density resulting from the simulation reports malignancy grade as a prognostic parameter. Moreover, our model permits to explore possible novel therapeutic procedures, including therapies targeting specific pathways. It captures cellular apoptosis by receptor inhibition in tumor development as a new area of mathematical modeling of targeted therapy.

Bladder cancer is a common type of cancer, and one with high recurrence rate and overall monitoring and treatment costs. It was developed and tested a three-dimensional agent-based computational model of the urothelium, a purpose-built Cellular Potts Model, to describe both a healthy urothelium and the initiation, development and invasion stages of a bladder cancer. This work is focused on the identification of the conditions in which tumor cells can cross, by mechanical means, the basement membrane and invade the bladder lamina propria. It is demonstrated how specific mechanical properties of the cancer cells, as their stiffness and adhesion to neighboring cells, heavily determine the probability of transition from a non-invasive to an invasive tumor. In particular, it is shown how an increased cellular adhesion of tumor cells to neighbor cells and/or extracellular fibers drives a collective migration, increasing the tumor growth rate and its invasion capacity. It is proposed that these physical characteristics should be considered as potential therapeutic targets to control the tumor growth and its progress to a more dangerous condition.

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.

Bladder cancer is one of the most common types of cancer, being the sixth more frequent in men, and one with higher recurrence rates and overall treatment costs. We introduce an agent-based computational model of the urothelium, adopting a Cellular Potts Model approach to describe both a healthy urothelium and the development of bladder cancer. We focus on the identification of the conditions in which cancer cells cross, by mechanical means, the basement membrane and invade the bladder lamina propria. When within the urothelium the tumor grows in a very constrained environment. These tight conditions imply that the urothelium layer where the tumor initiates greatly determines tumor growth and invasiveness. Moreover, we demonstrate how specific mechanical properties of the cancer cells, as their stiffness or the adhesion to neighboring cells, heavily modulate the critical initial moments of tumor development. We propose that these characteristics should be considered as therapeutic targets to control tumor growth. This article is protected by copyright. All rights reserved.

Keratins are one of the most abundant proteins in epithelial cells. They form a cytoskeletal filament network whose structural organization seriously conditions its function. Dynamic keratin particles and aggregates are often observed at the periphery of mutant keratinocytes related to the hereditary skin disorder epidermolysis bullosa simplex, which is due to mutations in keratins 5 and 14. To account for their emergence in mutant cells, we extended an existing mathematical model of keratin turnover in wild-type cells and developed a novel 2D phase-field model to predict the keratin distribution inside the cell. This model includes the turnover between soluble, particulate and filamentous keratin forms. We assumed that the mutation causes a slowdown in the assembly of an intermediate keratin phase into filaments, and demonstrated that this change is enough to account for the loss of keratin filaments in the cell&rsquo, s interior and the emergence of keratin particles at its periphery. The developed mathematical model is also particularly tailored to model the spatial distribution of keratins as the cell changes its shape.

In the summer of 2018, Professor Michael Kosterlitz visited Portugal as a plenary speaker of the FÍSICA2018 conference organised by the Portuguese Physical Society (SPF) and by the University of Beira Interior. FÍSICA 2018 comprised two meetings: the 28th Iberian Meeting for Physics Teaching, and the 21st National Conference of Physics, a biannual event that brings together researchers from all areas of Physics working in Portugal.

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., Graphical abstract fx1, Highlights • 2-Cys peroxiredoxins impose strong cytoplasmic and transmembrane H2O2 gradients. • Maximal cytoplasmic H2O2 concentrations are insufficient to oxidize Tyr phosphatases. • Prx inhibition modestly increases maximal H2O2 concentration, blunts gradients. • Sulfenate and disulfide Prx strongly exceed H2O2 concentrations, are also localized. • Prx recruitment causes modest local H2O2 depletion, helps Prx outcompete H2O2.

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.

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.

The formation of a functional vascular network depends on biological, chemical, and physical processes being extremely well coordinated. Among them, the mechanical properties of the extracellular matrix and cell adhesion are fundamental to achieve a functional network of endothelial cells, able to fully cover a required domain. By the use of a Cellular Potts Model and Finite Element Method it is shown that there exists a range of values of endothelial traction forces, cell-cell adhesion, and matrix rigidities where the network can spontaneously be formed, and its properties are characterized. We obtain the analytical relation that the minimum traction force required for cell network formation must obey. This minimum value for the traction force is approximately independent on the considered cell number and cell-cell adhesion. We quantify how these two parameters influence the morphology of the resulting networks (size and number of meshes).

Angiogenesis - the growth of new blood vessels from a pre-existing vasculature - is key in both physiological processes and on several pathological scenarios such as cancer progression or diabetic retinopathy. For the new vascular networks to be functional, it is required that the growing sprouts merge either with an existing functional mature vessel or with another growing sprout. This process is called anastomosis. We present a systematic 2D and 3D computational study of vessel growth in a tissue to address the capability of angiogenic factor gradients to drive anastomosis formation. We consider that these growth factors are produced only by tissue cells in hypoxia, i.e. until nearby vessels merge and become capable of carrying blood and irrigating their vicinity. We demonstrate that this increased production of angiogenic factors by hypoxic cells is able to promote vessel anastomoses events in both 2D and 3D. The simulations also verify that the morphology of these networks has an increased resilience toward variations in the endothelial cell’s proliferation and chemotactic response. The distribution of tissue cells and the concentration of the growth factors they produce are the major factors in determining the final morphology of the network.

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.

Um errata deste artigo encontra-se disponível em: http://hdl.handle.net/1822/52993, High Angular Resolution Diffusion Imaging (HARDI) is a type of brain imaging that collects a very large amount of data, and if many subjects are considered then it amounts to a big data framework (e.g., the human connectome project has 20 Terabytes of data). HARDI is also becoming increasingly relevant for clinical settings (e.g., detecting early cerebral ischemic changes in acute stroke, and in pre-clinical assessment of white matter-WM anatomy using tractography). Thus, this method is becoming a routine assessment in clinical settings. In such settings, the computation time is critical, and finding forms of reducing the processing time in high computation processes such as Diffusion Spectrum Imaging (DSI), a form of HARDI data, is very relevant to increase data-processing speed. Here we analyze a method for reducing the computation time of the dMRI-based axonal orientation distribution function h by using Monte Carlo sampling-based methods for voxel selection. Results evidenced a robust reduction in required data sampling of about 50 % without losing signal’s quality. Moreover, we show that the convergence to the correct value in this type of Monte Carlo HARDI/DSI data-processing has a linear improvement in data-processing speed of the ODF determination. Although further improvements are needed, our results represent a promissory step for future processing time reduction in big data., We thank the financial support by QREN, FEDER, COMPETE, Investigador FCT, FCT Ciencia 2007, FCT PTDC/SAU-BEB/100147/2008, FCT Project Scope UID/CEC/00319/2013, and the ERASMUS projects (FCT stands for "Fundacao para a Ciencia e Tecnologia"). We are thankful the relevant scientific conversations with Alard Roebroeck, Rainer Goebel, Van Wedeen, and Gina Caetano. Data collection for this work was in part from "Human Connectome Project" (HCP; Principal Investigators: Bruce Rosen, M.D., Ph.D., Arthur W. Toga, Ph.D., Van J. Weeden, MD). HCP funding was provided by the National Institute of Dental and Craniofacial Research (NIDCR), the National Institute of Mental Health (NIMH), and the National Institute of Neurological Disorders and Stroke (NINDS). HCP data are disseminated by the Laboratory of Neuro Imaging at the University of Southern California., info:eu-repo/semantics/publishedVersion

Tumor growth is becoming a central problem in biophysics both from its social and medical interest and, more fundamentally, because it is a remarkable example of an emergent complex system. Focusing on the description of the spatial and dynamical features of tumor growth, in this paper we review recent tumor modeling approaches using a technique borrowed from materials science: the phase-field models. These models allow us, with a large degree of generality, to identify the paramount mechanisms causing the uncontrolled growth of tumor cells as well as to propose new guidelines for experimentation both in simulation and in the laboratory. We finish by discussing open directions of research in phase-field modeling of tumor growth to catalyze the interest of physicists and mathematicians in this emergent field.

Systematic Monte Carlo simulations of simple lattice models show that the final stage of protein folding is an ordered process where native contacts get locked (i.e., the residues come into contact and remain in contact for the duration of the folding process) in a well-defined order. The detailed study of the folding dynamics of protein-like sequences designed as to exhibit different contact energy distributions, as well as different degrees of sequence optimization (i.e., participation of non-native interactions in the folding process), reveals significant differences in the corresponding locking scenarios—the collection of native contacts and their average locking times, which are largely ascribable to the dynamics of non-native contacts. Furthermore, strong evidence for a positive role played by non-native contacts at an early folding stage was also found. Interestingly, for topologically simple target structures, a positive interplay between native and non-native contacts is observed also toward the end of the folding process, suggesting that non-native contacts may indeed affect the overall folding process. For target models exhibiting clear two-state kinetics, the relation between the nucleation mechanism of folding and the locking scenario is investigated. Our results suggest that the stabilization of the folding transition state can be achieved through the establishment of a very small network of native contacts that are the first to lock during the folding process.

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, that 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.

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.

We study the effect of a uniform shear flow on an interface separating the two broken-symmetry ordered phases of a two-dimensional system with nonconserved scalar order parameter. The interface, initially flat and perpendicular to the flow, is distorted by the shear flow. We show that there is a critical shear rate, \gamma_c, proportional to 1/L^2, (where L is the system width perpendicular to the flow) below which the interface can sustain the shear. In this regime the countermotion of the interface under its curvature balances the shear flow, and the stretched interface stabilizes into a time-independent shape whose form we determine analytically. For \gamma > \gamma_c, the interface acquires a non-zero velocity, whose profile is shown to reach a time-independent limit which we determine exactly. The analytical results are checked by numerical integration of the equations of motion., Comment: 5 pages

Recent work has suggested the existence of glassy behavior in a ferromagnetic model with a four-spin interaction. Motivated by these findings, we have studied the dynamics of this model using Monte Carlo simulations with particular attention being paid to two-time quantities. We find that the system shares many features in common with glass forming liquids. In particular, the model exhibits: (i) a very long-lived metastable state, (ii) autocorrelation functions that show stretched exponential relaxation, (iii) a non-equilibrium timescale that appears to diverge at a well defined temperature, and (iv) low temperature aging behaviour characteristic of glasses.

In the phase separation occurring at the miscibility gap (at the spinodal region) of an alloy a discrete symmetry is spontaneously broken and a domain wall network is formed. Field theory simulations are often used to study the dynamics of topological defects networks appearing in different physical contexts. In this work, we focus on the dynamics of the two immiscible liquids appearing on the phase diagram of the Bi–Zn system, one of the basic systems of lead free solders. We use phase field simulations to quantitatively simulate the dynamics of the two liquids separation in the Bi–Zn system, at different temperatures and for different concentrations. We obtain the miscibility gap curve and the domain morphologies of the system as a function of time, temperature and component concentrations using simulations.

Redundancy constitutes a fundamental and intrinsic aspect of healthy vasculatures. Built-in redundancy might also be a desirable feature in man-made microfluidic devices. We show that redundant and non-redundant networks, built to have identical resistances to flow when unobstructed, allow for very different flows when they are occluded; redundant ones??densely occluded at a certain bifurcation level??allowing for larger flows than non-redundant ones??obstructed above relatively small thresholds. We also show that redundancy protects vessels against the large shear-rate gradients that occlusions would cause if it were not present. Our study allows one to quantify a network tolerance against blockage, provides guidance in the tailoring of microfluidic devices, and offers novel insights into why nature has selected intrinsic redundancy over thicker vessels to assure blood supply at key places of the organisms.

The purpose of this work is to develop improvements on the most modern forms of white matter fiber tracking in the human brain. This development requires the use of computer simulations to assess the limits of the fiber tracking technology, and then the results of that simulation are used to make a thorough assessment of the experimental results in living human brains. The used fiber tracking techniques are q-ball imaging and diffusion spectrum imaging.

A simulation of a hybrid multi-scale phase-field model which describes the dynamics of the interface that divides the new capillaries and the stroma was performed. Four equations model the behaviour of proliferating endothelial cells (ECs) in response to a gradient of a pro-angiogenic factor (T), leading to the growth and time evolution of a tree of vascular vessels from sprouts emerging from the original capillar. New sprouts are pulled by activated tip cells and move with a velocity proportional to the gradient of angiogenic factors. The sources of pro-angiogenic factor are hypoxic cells. This factor diffuses throughout a three-dimensional system and is consumed by the ECs whose proliferation is governed by an order parameter (φ). Higher values of chemotactic response led to an increase of the number of ramifications in the vessels, which were thinner; higher values of proliferation rate to thicker vessels and a more ramified network and an increase in the number of hypoxic cells led to a higher number of branches, higher angiogenic factor consumption, longer and internally more proliferative vessels.

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.

The process of sprouting angiogenesis is extremely complex involving hundreds of proteins that regulate transcription and participate in signaling pathways controlling cellular movement, proliferation and phenotype alteration. Modeling has been attempted to understand all these mechanisms, and hence, in this chapter, we will focus on models that deal individually with each one of these mechanisms relevant to angiogenesis, as well as with platforms that integrate various models into multiscale models of the whole process.

Blood vessel growth is pivotal in various processes in health and disease; examples are embryogenesis, inflammation, wound healing, cardiac ischemia, diabetic retinopathy, and tumor growth. The latter, in particular, has been the subject of extensive studies due to the high mortality rates associated with oncologic diseases. However, and despite the continuous efforts by the scientific community, both the genomic characterization of tumors (Maley et al., 2006; Wood et al., 2007) and the main physical mechanisms driving their development (Araujo & McElwain, 2004) are currently a matter of debate. These efforts have been greatly enriched through an interdisciplinary approach. Physics, Mathematics and computer simulation and modeling have currently a key role in the research of tumor growth in general and vessel development in particular. The approach of mathematicians and physicists provides remarkable new ways of looking into Biology: starting the modeling from basic principles, the fundamentals of the problem can be tested and understood. The relevance of blood vessel growth for tumor development is well documented (Figg & Folkman, 2008). In the early stages of development, the growth of a solid tumor is limited in size due to cell apoptosis at its core by lack of nutrients. In simple terms, the acquisition of nutrients at the boundary is not able to meet the needs of the inner cells. This, in addition to a problem of confinement of the inner cells (the fast growth does not allow them to migrate) produces the so called core necrosis inside the tumor. When this stationary non-threatening stage is reached, malignant tumors adapt to the environment and undergo an active search for the nutrients they need to survive and proliferate. This can be achieved by releasing factors, such the Vascular Endothelial Growth Factor (VEGF), which drive nearby vessels to extend new branches in the direction of the tumor, through a process called angiogenesis, providing it with nutrients (Figg & Folkman, 2008). Hence, understanding angiogenesis is essential to the potential control of the blood delivered to a neoplasic tissue in order to prevent and contain its development and metastatic colonies. In fact, though hyper-vascularization is not a requirement for a tumor to metastasize (Mateus et al., 2009), the cells of a densely vascularized tumor have a much higher facility in entering the blood circulation. During the last decade cancer therapies based on anti-angiogenic factors have been a focus of interest. However, this type of treatment has not given the expected results (Mayer, 2004) and recently, the concept of normalization of tumor vasculature has been proposed (Jain, 2005), where it is suggested that certain anti-angiogenic agents can transiently normalize 9

Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

Systematic Monte Carlo simulations of simple lattice models show that the final stage of protein folding is an ordered process where native contacts get locked (i.e., the residues come into contact and remain in contact for the duration of the folding process) in a well-defined order. The detailed study of the folding dynamics of protein-like sequences designed as to exhibit different contact energy distributions, as well as different degrees of sequence optimization (i.e., participation of non-native interactions in the folding process), reveals significant differences in the corresponding locking scenarios--the collection of native contacts and their average locking times, which are largely ascribable to the dynamics of non-native contacts. Furthermore, strong evidence for a positive role played by non-native contacts at an early folding stage was also found. Interestingly, for topologically simple target structures, a positive interplay between native and non-native contacts is observed also toward the end of the folding process, suggesting that non-native contacts may indeed affect the overall folding process. For target models exhibiting clear two-state kinetics, the relation between the nucleation mechanism of folding and the locking scenario is investigated. Our results suggest that the stabilization of the folding transition state can be achieved through the establishment of a very small network of native contacts that are the first to lock during the folding process.

We perform extensive lattice Monte Carlo simulations of protein folding to construct and compare the equilibrium and the kinetic transition state ensembles of a model protein that folds to the native state with two-state kinetics. The kinetic definition of the transition state is based on the folding probability analysis method, and therefore on the selection of conformations with 0.4

We apply a simulational proxy of the phi-value analysis and perform extensive mutagenesis experiments to identify the nucleating residues in the folding reactions of two small lattice Go polymers with different native geometries. These results are compared with those obtained from an accurate analysis based on the reaction coordinate folding probability Pfold, and on structural clustering methods. For both protein models, the transition state ensemble is rather heterogeneous and splits-up into structurally different populations. For the more complex geometry the identified subpopulations are actually structurally disjoint. For the less complex native geometry we found a broad transition state with microscopic heterogeneity. For both geometries, the identification of the folding nucleus via the Pfold analysis agrees with the identification of the folding nucleus carried out with the phi-value analysis. For the most complex geometry, however, the apllied methodologies give more consistent results than for the more local geometry. The study of the transition state' structure reveals that the nucleus residues are not necessarily fully native in the transition state. Indeed, it is only for the more complex geometry that two of the five critical residues show a considerably high probability of having all its native bonds formed in the transition state. Therefore, one concludes that in general the phi-value correlates with the acceleration/deceleration of folding induced by mutation, rather than with the degree of nativeness of the transition state, and that the traditional interpretation of phi-values may provide a more realistic picture of the structure of the transition state only for more complex native geometries., Comment: Submitted for publication with minor changes (abstract & higher resolution figures)

We perform extensive Monte Carlo simulations of a lattice model and the Go potential to investigate the existence of folding pathways at the level of contact cluster formation for two native structures with markedly different geometries. Our analysis of folding pathways revealed a common underlying folding mechanism, based on nucleation phenomena, for both protein models. However, folding to the more complex geometry (i.e. that with more non-local contacts) is driven by a folding nucleus whose geometric traits more closely resemble those of the native fold. For this geometry folding is clearly a more cooperative process., Comment: Accepted in J. Chem. Phys

Using computer simulations, we establish an approach for creating defect-free, periodically ordered polymeric materials. The system involves ABC ternary mixtures where the A and B components undergo a reversible photochemical reaction. In addition, all three components are mutually immiscible and undergo phase separation. Through the simulations, we model the effects of illuminating a three-dimensional (3D) sample with spatially and temporally dependent light irradiation. Experimentally, this situation can be achieved by utilizing both a uniform background light and a spatially localized, higher intensity light, and then rastering a higher-intensity light over the 3D sample. We first focus on the case where the higher-intensity light is held stationary and focused in a distinct region within the system. The C component is seen to displace the A and B within this region and replicate the pattern formed by the higher-intensity light. In effect, one can write a pattern of C onto the AB binary system by focusing the higher-intensity light in the desired arrangement. We isolate the conditions that are necessary for producing clearly written patterns of C (i.e., for obtaining sharp interfaces between the C and A/B domains). We next consider the effect of rastering a higher-intensity light over this sample and find that this light "combs out" defects in the AB blend as it moves through the system. The resulting material displays a defect-free structure that encompasses both a periodic ordering of the A and B domains and a well-defined motif of C. In this manner, one can create hierarchically patterned materials that exhibit periodicity over two distinct length scales. The approach is fully reversible, noninvasive, and points to a novel means of patterning with homopolymers, which normally do not self-assemble into periodic structures.

Computer simulations reveal how photoinduced chemical reactions can be exploited to create long-range order in binary and ternary polymeric materials. The process is initiated by shining a spatially uniform light over a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and phase separation. We then introduce a well-collimated, higher intensity light source. Rastering this secondary light over the sample locally increases the reaction rate and causes formation of defect-free, spatially periodic structures. These binary structures resemble either the lamellar or hexagonal phases of microphase-separated diblock copolymers. We measure the regularity of the ordered structures as a function of the relative reaction rates for different values of the rastering speed and determine the optimal conditions for creating defect-free structures in the binary systems. We then add a nonreactive homopolymer C, which is immiscible with both A and B. We show that this component migrates to regions that are illuminated by the secondary, higher intensity light, allowing us to effectively write a pattern of C onto the AB film. Rastering over the ternary blend with this collimated light now leads to hierarchically ordered patterns of A, B, and C. The findings point to a facile, nonintrusive process for manufacturing high quality polymeric devices in a low-cost, efficient manner.

Computer simulations reveal how photoinduced chemical reactions in polymeric mixtures can be exploited to create long-range order in materials with features that range from the submicron to the nanoscale. The process is initiated by shining a spatially uniform light on a photosensitive AB binary blend, which thereby undergoes both a reversible chemical reaction and a phase separation. When a well-collimated, higher intensity light is rastered over the sample, the system forms defect-free, spatially periodic structures. If a nonreactive homopolymer C is added to the system, this component localizes in regions that are irradiated with a higher intensity light, and one can effectively "write" a pattern of C onto the AB film. Rastering over the ternary blend with the collimated light now leads to hierarchically ordered patterns of A, B, and C. Because our approach involves homopolymers, it significantly expands the range of materials that can be fashioned into a periodic pattern. The findings point to a facile process for manufacturing high-quality polymeric components in an efficient manner.

Through a combination of simulation techniques, we determine both the structural evolution and mechanical properties of blends formed from immiscible ternary mixtures. In this approach, we first use the lattice Boltzmann method to simulate the phase separation dynamics of A/B/C fluid mixtures for varying compositions within the spinodal region. We also investigate the effect of an imposed shear on the phase ordering of the mixture. We assume that the fluid is quenched sufficiently rapidly that the phase-separated structure is preserved in the resultant solid. Then, the output from our morphological studies serves as the input to the lattice spring model, which is used to simulate the elastic response of solids to an applied deformation. These simulations reveal how the local stress and strain fields and the global Young's modulus depend on the composition of the blend and the stiffness of the components. By comparing the results for the sheared and unsheared cases, we can isolate optimal processing conditions for enhancing the mechanical performance of the blends. Overall, the findings provide fundamental insight into the relationship between structure, processing, and properties for heterogeneous materials and can yield guidelines for formulating blends with the desired macroscopic mechanical behavior.

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 vesseldamage. Based on this property of vascular networks, we propose a novel mechanism for the transition between the nonproliferative and the roliferative stages in diabetic retinopathy.

We consider the interplay of thermal fluctuations and shear on the surface of the domains in various systems coarsening under an imposed shear flow. These include systems with nonconserved and conserved dynamics, and a conserved order parameter advected by a fluid whose velocity field satisfies the Navier-Stokes equation. In each case the equation of motion for the interface height reduces to an anisotropic Burgers equation. The scaling exponents that describe the growth and coarsening of the interface are calculated exactly in any dimension in the case of conserved and nonconserved dynamics. For a fluid-advected conserved order parameter we determine the exponents, but we are unable to build a consistent perturbative expansion to support their validity., Comment: 10 RevTeX pages, 2 eps figures