Paulo Casagrande Godolphim
Paper 4: Tumor Aggressiveness Relates to Jamming Dynamics in Epithelial Monolayers
Main Objective (Idea to introduce, Question to Answer)
- There is a relation between collective movement profile and tumor aggressiveness in epithelial monolayers.
- The already know features associated with pos-EMT metastatic cells can physically explain these profiles (via simulation and theory)?
Sobre esses dois primeiros itens: dependedo da adesão entre as células pode ou não ocorrer congelamento abrupto da dinâmica (celular jamming), com mais ligações, ou ligações mais fortes, ou pathways mais eficientes, o comprimento de correlacao fica maior, o que provaco o congelamento dos modos de oscilaçao maiores, resultanod nesse perfil abrupto de congelamento das velocidades. Já para a célula metastatica, as ligaç~oes suposta,emte mais fracas, teria assim comprimento de correlação menor, não apresentando nenhuma queda abrupta.
- Tumor cells monolayers undergo jamming transition as like healthy cells? If not, what type of transition they are (what is the nature of the transition, could be a coagling transition)?
- Corroborate with Garcia 2015 idea that cellular jamming is controlled by junctions maturation, rather than density.
- But to show that density still play a role associated to the collective movement intensity, and by so, to the severity of the velocity arrest during transition.
- Mount top pipe-line cancer analises assay, as done by PS-ON 2013), but easy reproducible, inexpensive, stathistacally robuts ("physical soul") and the closest possible to clinic.
- And if the cotutela vai ser perseguida, montar esse projeto de modo vendável com as devidas perspectivas (essa linha não ficara aqui e ira para perspectivas dai).
Cell Density Heterogeneity
How density is distributed along the monolayer is an opened question. We need to know this because of four things:
(1) We belive to be Garcia et al 2015 via figura A, because we separate it vi time well. But lokking to B we se that the initial densities are also diferent and since we only have one field of view for each curve, maybe this is related with the density, and not to time. We will only be sure of that if we determine how is the density distribution along the monolayer. E o proprio trabalho do garcia mostra ginates fllutuacoes que sugereriam grande flutuacoes.
(2) We thinks density still play a role by modulating the collectiv movimment intensity, as shown by figure C. So it would be very apropriated to know how is density distribution, since it is related with the intenisty of moviment. For example, it could be not the absolute density on the field of view that really matters (like we are fiding until now), but how heterogenius. This heterogenity would allow compression and "free space" to the cells move and then creae this velocity profiles.
(3) One of the main objectives is to produce a experimental/analizes pipeline that is rbust and easy to do. This pipe line should not ignorate a relevant subject as density fluctuation, unles it is comproved that is irrelevant.
(4) In the "work perspectives" scenario, this information of diference in heterogenity could be a very important analizes tool in the comparison of cancer monolayers in future studies and so...
So what can we do about it? There are two approaches:
(1) We could do a cross grain and faine graind density check along the monolayer, as shown by figure 2. This would be a very trourogh examination, but very time consuming and data generator.
(2) Or we could do local analyzes and the use of good physics. Garcia et al 2015 found that a monolayer of human bronchial epithelial cells (HBEC) present giant density (number) fluctuations (see figure Garcia-2015-figS1), so maybe we could just do that and will not need to be inspecting the whole monolayers. Garcia did that citing this two works Duclos et al 2014, Marchetti et al 2013, and the counting was with flourescente cells microscoppy and the analizes ans segmentation was with fiji and they cited the main fiji work Schindelin et al 2012. We will use garcia stes as a guide. What we have to do is to count the number of cells inside the FOV, save this marked file as justa the marks, use some plugin on fiji (if ther is one) to then segmentait the image e count just like Garcia did.
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Garcia Vrms monotonic decay, natural units, and stuff...
Section to talk about of: We say that SCC9 has a monotonic decay, while Hacat has a characteristic sharp decay. This line of artgumentation that we use to introduce taht the malignat one has a diference and then the relation fo the principal result of the work. But with look Garcia 2015 he say that the velocity drop of his cells (hralty one and very similar to hacat) has a mnotonic decay. This , a priori dont devalidate our result, but we ned to take a look in that to "falar um descurso comum". But One thing that could, maybe, put to zero our results is the natural units. As giberto, rita and Statphys japa said: I can't be sure taht this curves are diferents really, if they are not in natural unitis.
Personally I belive every thing will be okay after doing that and our conclusion will be that the more discrete deacres in velocity of SCC9 is related to they nao poderem reach hihg speeds after wall, because of they week connection. If no high speed, no hard break. Simple as that.
Deadlock on Monolayers Velocity Correlation
First comment: We don't need to solve this deadlock for our research to go on, but is important that we know it exists and the possibles causes for it.
Hello good by question: What other works are telling about velocity correlation on monolayers?
Two main references for our work seams to present distinct velocity correlation profiles. While Angelini et al 2010 shows that the velocity correlation function have a well defined minimum (that they use to define the swirls correlation length; graph C of Angelini-2010-fig3), Garcia et al 2015 show a exponential fitted decay with no critical point (graph C of Garcia-2015-fig1). Why this, if same equation-link equation-link. Maybe substrate, cell linage, whats is happening. Finisinh wit MOSAIC-COmparison.
Methodology
This section is not to show results, but to show how things are being done. In other words, it is a methodology section (supriiise!). It start with short overview of all the work pipeline (flowchart), and then a detail explanation of each step. The motivation for doing each step is not the focus of this section, so there are just short comments about it. When a thorough explanation were judged necessary, it was done in separated sections. For example in the case of cell density heterogeneity, where the explanation section can be found here.
Pipe Line Overview (Flowchart)
The link below show an overview of the work pipeline with a flowchart. This image is a print screen for the methodology section from my Statphys 2019 poster. A better image will be made for this section, along with better explanations. media:wiki-mosaic4.png
Experiments and Data Acquisition
Image Treatments and Measurements (Image Analysis)
In this image analysis section we talk about image treatments and analysis techniques employed in each measurements. The measurements are explained (along with their limitations) and the quality of the resulting data is exposed.
- Velocity field
- Number of cells
- Specific proteins concentrations
Computed Quantities
- Velocity correlations
Velocity correlation is calculated over a velocity field and give information of a typical ordering length (like in a collective movement approach), or information propagation length, or clusters size. Probably the first use (or more famous use) of it in monolayers quantification was done by Angelini et al 2010, where they used the correlation length to show how the swirl characteristic size changed in time for a growing MDCK epithelial monolayer (they were probably, I think, persuing an idea of universality class style, just like in Vicsck school of thought). Garcia et al 2015 found that a monolayer of human bronchial epithelial cells (HBEC) present giant density (number) fluctuations (see figure Garcia-2015-eqS18)
- Mean velocities (Vrms, activity, drift, heat map, fluctuations, energy waste)
- Density (giant number fluctuations, growth rate)
Simulation and Theory
(partial) Results
As a first comment it is very important to dig in Garcia et al 2015. This work is a manual for us. and to do more experiments and simulations (and more literature reviews).
Title result - Tumor aggressiveness relates to jamming dynamics in epitelial monolayers
Alternative title: Collective velocity profile during epithelial solidification relates to oral cell linage malignancy on in vitro monolayer experiments.
Why this alternative title: To be more precise (stressing our limitations) and to not talk about jamming.
Why not talk about jamming: taking a good look into Garcia et al 2015, they put the word jamming on the title, but they avoid talking about a jamming transition, instead, referring to it as solidification process and glassy dynamics. The reasons is that they, just like us, don't really know the nature of this process, and so, generically referring to it as a solidification process that resembles glassy dynamics. Falar berthier point and references of jamming vs glass tranbsition and maybe talk about Meanning jamming experiments.
Figure 3 - Aggressiveness relates to jamming dynamics
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Literature Review
This is a literature review focused on understand specific topics related with the work. The Idea is to start with one or two main references for each topic and enlarge it by diving on their on references.
Physics of cancer
This review have three main objectives:
- How novel really is what we are proposing.
- Compare the methodology of article (in ~9.0 Impact Factor journals) with ours.
- Find articles that give us a hint of how far from clinical implementation we are.
- White 2019, The Physics of Cancer, AACR
- Review article
- About: It's a review of what has been done in the first decade of the Physical Science-Oncology Network (PS-ON) experiment, supported by the US National Cancer Institute (NCI).
- Contribution: "... highlight key areas in which the two disciplines have been successfully integrated ..."
- Limitation/bias: It only reviews PS-ON articles, so this is a inside the US bubble "Physics of cancer" review. It's a big bubble, but a bubble yet.
- Found: Google.
- Lessons learned for our work:
- Microenvironment is known to be a key factor in tumor growth.
- The idea of using physical measures direct in to clinical world is strongly defended here. So our idea, in it's broad way, is not a novelty. That is good, but they are talking about having "on the fly" (minute to second) in situ tracking of tumor responses to treatments, able doctors to adjust treatment rapidly making the tumor unable to adjust/self-select fast enough and then not reach critical point/mass/state. This idea is much more sophisticated and multidisciplinary, the physics enter in producing non-invasive equipment that can analyze this systems, give response and wit the all ready know influence of physical relations and malignancy, to the treatment. I think our idea can be used in the benefit of this main idea.
- PS-ON 2013, A physical sciences network characterization of non-tumorigenic and metastatic cells, Scientific Reports
- Human epithelial cells, linage comparison, non-tumorigenic to metastatic, single cell analyses, biophysics
- About: Team effort of 12 research groups (from PS-ON collaboration)
- Contribution: Show that is possible to large colaboration in cancer if have stadarizided protocls assays.
- Limitation/bias:
- Found: White 2019, AACR
- Lessons learned for our work:
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Jamming in cancer (and in medical applications)
This review have two main objectives:
- To know the articles that did monolayer cancer experiments: if they talk about jamming, what are their methodology/analyzes, and conclusions.
- How Jamming theories are (and if are) entering in the medical/clinical/drug/treatment (not just cancer) world? Find references of it.
- Oswald 2017, Jamming transitions in cancer, JPD:AP
- Review article
- About: A review about jamming on cancer: need to read again!
- Contribution:
- Limitation/bias:
- Found: Google.
- Lessons learned for our work:
- ...
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Cellular jamming
This review have X main objectives:
- How novel really is something ...
- ...
- Garcia 2015, Physics of active jamming during collective cellular motion in a monolayer, PNAS
- Human epithelial cell, single linage, junction maturation vs density, monolayer analyses, biological physics
- About: I wrote "single linage", but actually then show that the same frame work works for other two lineages as well.
- Contribution: Propose that cells junctions maturation, and not density, controls jamming transition in epithelial monolayers. This last frase is not right, just because I used the term "jamming transition". They put jamming in the title, but they never refer to this transitions in an apropriated context. The talk about glass transition, they are cautious about talking "jamming transition". We have to do the same.
- Limitation/bias:
- Found: Google.
- Lessons learned for our work:
- ...
- The first cluster annalizes is only to justify the Correnlention lehgt vs Vrms curve and to give physical interpretaions/predictions.
- The efecive measuremnt of cluster is used just in the part to acess dynamical hetrogeniety, by tracking the fasted cluster on the FOV
- Angelini 2010, Cell Migration Driven by Cooperative Substrate Deformation Patterns, PRL
- Topic 1, topic 2, ... - Specific sub areas of the article
- About: A seminal work on epithelial monolayer physics. It's not actually centered on jamming (or glass like solidification), but is about a growing monolayer where quantities are plotted vs density increase.
- Contribution: "quantify spatial and temporal correlations in migration velocity and substrate deformation, and show that cooperative cell-driven patterns of substrate deformation mediate long-distance mechanical coupling between cells and control collective cell migration."
- Limitation/bias: It seams that they stop the analyses before the solidification transition. This is not a problem for their results, but is important that WE know this when using does results. This is a very good and important work, but maybe it has a too much phisicist naive aproach when interpeting the biological results (see article by it self section (link)).
- Found: Helene showed me.
- Lessons learned for our work:
- Motivated the beginning of this work in collaboration with Lamoc.
- Use velocity autocorrelation function minima to define the swirl characteristic length (see angelini-2010-eq1 and angelini-2010-fig3).
Image Analysis
This review have the main objective to save/collect articles about:
- Fiji (image treatment, cell counting, automation)
- OpenPIV (or any PIV information)
- AI (cell identification and counting)
- Maybe KLT and others cell/object tracking softwares
- Schindelin 2012, Fiji: an open-source platform for biological-image analysis, Nature Methods
- Fiji, open software release
- About: 15k citations!!!!! From Max Plank of Dresden, Germany.
- Contribution: Sheared the open source version of Image J
- Limitation/bias:
- Found: Garcia 2015
- Lessons learned for our work:
- It is the "Know me" article of Fiji, the one to reference it
Active Matter
This review have objectives:
- The objectives are specific, each paper added by a singular reason, like demostrate a method in dethail
- Marchetti 2013, Hydrodynamics of soft active matter, Reviews of Modern Physics
- Review article, 47 páginas!!!!
- About: It is as 47 pages review in active matter!
- Contribution: I don't know, added just because Garcia 2015 cited this work as a reference for giant number fluctuations.
- Limitation/bias:
- Found: Garcia 2015
- Lessons learned for our work:
- Giant number fluctuations: Garcia 2015 cited this work as a reference for this subject.
- Duclos 2014, Perfect nematic order in confined monolayers of spindle-shaped cells, Soft Matter
- Mouse fibroblast cells, constrained monolayer, glass transition, disorder to nematic order, physics
- About: This is a physics paper "by desing", but is about almost apolar (zero adehsion) fibroblast cells growing as a cross constrained monolayer, presenting disorder to order nematic alighment and annomulous density fluctuations.
- Contribution: Not so sure, added this articles because Garcia 2015 cited it as a reference for Giant number fluctuations. But I thinks they show that fibroblast have anomulus density fluctuation and other interesting things.
- Limitation/bias:
- Found: Garcia 2015
- Lessons learned for our work:
- Have a entire section about giant number fluctuations.
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Broad Area - Sample Templates
This review have X main objectives:
- How novel really is something ...
- ...
- Name year, Title, Journal - Sample Template
- Topic 1, topic 2, ... - Specific sub areas of the article
- About:
- Contribution:
- Limitation/bias:
- Found:
- Lessons learned for our work:
- ...
- ...
Articles by themselves (in alphabetical order)
Angelini 2010, Cell Migration Driven by Cooperative Substrate Deformation Patterns, PRL
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Naive "too much physicist" interpretation, but interesting even so:
- "At lower densities, cell shapes fluctuate dramatically, whereas at higher densities, cell shapes change little and remain nearly round. Thus, smaller rounder cells coordinate motion in larger numbers of cells and over greater distances than do the larger more dynamic cells." This boldface "thus" represents the type of "physicist doing biology" jumped conclusion we want to avoid. For more context, search for a red note in the article file on Mendeley library.
- Figures, Graphs and Movies:
Duclos 2014, Perfect nematic order in confined monolayers of spindle-shaped cells, Soft Matter
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
- ...
Garcia 2015, Physics of active jamming during collective cellular motion in a monolayer, PNAS
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
Marchetti 2013, Hydrodynamics of soft active matter, Reviews of Modern Physics
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
- ...
PS-ON 2013, A physical sciences network characterization of non-tumorigenic and metastatic cells, Scientific Reports
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
- ...
Schindelin 2012, Fiji: an open-source platform for biological-image analysis, Nature Methods
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
- ...
Oswald 2017, Jamming transitions in cancer, JPD:AP
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
- ...
White 2019, The Physics of Cancer, AACR
- Introduction:
- "The societal and personal burden of cancer has stimulated decades of intense scientific effort that has resulted in many important insights and therapies"
- "Yet, the improvement in mortality rates for patients with cancer still lags behind that of cardiovascular and cerebrovascular diseases."
- "Research has been greatly accelerated by new experimental technologies and the revolution in genomics and bioinformatics."
- "These new methodologies have generated overwhelming amounts of biomolecular data, but lacking the conceptual frameworks necessary to organize it ..."
- "the common focus on genes and gene products .. often neglects the physical context in which clinical cancer cells grow."
- "Yet, all stages of cancer are impacted by the 3-dimensional (3D) microenvironment ... subject to complex mechanical forces and spatiotemporally varying gradients of biomolecules and nonorganic components such as oxygen and acid."
- "Cancer cells can also deploy "niche construction" strategies, including extracellular matrix (ECM) remodeling, angiogenesis, and extracellular acidification, to make their microenvironment permissive to tumorigenesis."
- Since 2009 the PS-ON study cancer, focusing in two (created?) areas (1) Physical Dynamics of Cancer and (2) Spatio-Temporal Organization and Information Transfer in Cancer. *" investigate the role of physical forces and microenvironmental factors in cancer, the PS-ON complements the Cancer Systems Biology Consortium of the NCI, which addresses challenges of cancer complexity by combining experi- mental biology with in silico modeling, multidimensional data analysis, and systems engineering."
- "one decade into this multidisciplinary experiment, several "lessons learned" and two general areas have emerged."
- "One broad topic applies physical sciences techniques to measure key biomechanical forces that affect cancers across multiple length scales...molecules,cells,tumortissue." :*"These properties are important causes and consequences of malignant transformation and tumor growth"
- "A second broad topic embraces the physics research paradigm, dating back to Newton and Galileo, in which mathematically based theoreticians work closely with experimentalists to define the first principles of a system."
- "Below, we highlight key accomplishments and outstanding opportunities in these specific areas."
- Physical Properties of Cancer Cells and Their Microenvironment:
- Cancer is naturally accepted as a gene related disease and even though that "the observation that tumors are stiffer than normal tissues enabled cancer diagnosis by palpation for centuries; and that tumors exhibit aberrant transport properties and, thus, metabolize nutrients differently than their normal counterparts has contributed to Otto Warburg's Nobel Prize in Physiology or Medicine in 1931. That varied tissue mechanics and transport properties can independently stimulate malignant transformation has only become clear relatively recently due, in part, to pioneering work by the PS-ON (1)."
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Name year, Title, Journal - Sample Template
- Introduction, Results, Discussion/Conclusions, Methodology:
- ...
- Figures, Graphs and Movies:
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Papers Proposals
Paper 5 - Can melanocytes nematic phase nuclear alignment be explained only by steric effects?
Projeto: Experimento in silico de matéria ativa sujeita a restrições geométricas - Projeto de Iniciação Científica
O ramo da física Matéria Ativa começou, para muitos, com o trabalho de 1995 de Vicsek et al. [1] onde foi introduzido um modelo mínimo de partículas autopropelentes sujeitas somente a interações simples e locais. O modelo é capaz de descrever comportamentos emergentes (de movimento coletivo sem liderança) bem como a transição de fase entre o movimento coletivo desordenado e ordenando, propriedades essas características de uma gama de sistemas complexos biológicos e não biológicos.
Baseado na estrutura de boids (bird-oid objects) de Craig Raynolds 1986 [2], o modelo de Vicsek desenvolve a dinâmica de partículas livres puntiformes interagindo entre si dentro de uma caixa 2D finita com contorno periódico e com passo de velocidade de módulo constante sujeitas somente a duas simples interações: uma que tende a alinhar (ordenar) a velocidade da partícula com as velocidades de suas vizinhas - dentro de um raio crítico de interação - e um ruído branco intrínseco à velocidade das partículas que tende a desalinhar (desordenar) as mesmas. Como o módulo da velocidade é mantido sempre constante, as interações no sistema atuam somente nas posições angulares dos vetores velocidade.
O grande sucesso do modelo se dá ao fato de que, apesar da evidente simplicidade, ele apresenta a emergência de comportamento coletivo ordenado que foi (e ainda é) usado para descrever o comportamento de sistemas complexos e altamente fora do equilíbrio como o movimento de revoadas, cardumes e até o crescimento de bactérias . Assim é possível estudar e inferir características de sistemas cuja a experimentação in vivo e in vitro é muito custosa (tanto financeiramente como laboralmente), valendo-se de experimentos in silico de baixo custo.
Desde o sucesso do modelo, várias modificações foram apresentadas criando uma classe de modelos tipo Vicsek, nos quais é conservada a estrutura geral modificando somente determinadas características como o tipo de contorno, de interações, as regras de vizinhos. Em 2003 Grégoire et al. introduziram uma interação radial, derivada de um potencial harmônico, possibilitando a coesão do sistema mesmo para um limite de densidade zero e um volume de exclusão (esfera rígida), impedindo que haja sobreposicão e cruzamento entre partículas [3]. Tais modificações construíram um modelo mais fidedigno às características de certos sistemas, tendo sido usado para descrever o movimento de amebas de dicty [7], segregação celular [4],[5]. Com o modelo de Vicsek-Grégoire é possível obter não somente a transição ordem-desordem, mas também transições de fase do tipo sólido-líquido-gasoso (com ou sem movimento). De certo modo, diríamos que o modelo de Grégoire está para o modelo de Vicsek, assim como o modelo de Van der Waals está para o gás ideal.
Em 2011 Vedula et al. [6] realizaram um experimento in vitro sobre migração celular (de tecidos) onde folhas de células epiteliais MDCK - Mandin-Darbi canine kidney - são liberadas de um reservatório para trilhos de fibronectina de larguras distintas (do diâmetro de uma célula até dezenas), porém pequenas em relação a largura do reservatório. Comportamentos emergentes coletivos distintos foram identificados quando variou-se as larguras dos trilhos: nos trilhos largos foram identificados vórtices de dezenas de células de comprimento, nos trilhos finos as células apresentaram um movimento como o de lagartas (caterpillar like) e que quanto mais finos os trilhos, maior a velocidade de migração das células. Tais respostas às restrições geométricas levantaram hipóteses de como as interações microscópicas entre as células estariam conduzindo o movimento macroscópico coletivo das mesmas. Utilizaremos o modelo mínimo de Vicsek-Grégorie para realizar experimentos in silico afim de testar hipóteses de como as interações célula-célula estariam relacionadas com os padrões de migração coletivos quando sujeitas a restrições geométricas do modelo de engarrafamento.
Metodologia
A revisão bibliográfica será feita durante todo o projeto e a escrita do relatório será delegada aos últimos meses do projeto.
Começaremos desenvolvendo um programa em FORTRAN para aplicar o modelo de Vicsek-Grégoire e garantir que o nosso programa esteja apresentando resultados de acordo com a literatura (as características a serem medidas e comparadas com a literatura ainda serão definidas). Em paralelo estudaremos a fundo o artigo de Vedula [6] afim de definir quais seriam as grandezas de interesse a serem medidas (por exemplo, as grandezas que possibilitariam identificar os movimentos caterpillar like ou os vórtices).
O passo seguinte será desenvolver o programa, sem otimização, afim de obter intuição sobre como as variações dos parâmetros induzem características de movimento distintas. Utilizando condição de contorno aperiódica do tipo de parede rígida, construiremos um reservatório retangular com capacidade de armazenar centenas de células. Ao reservatório será conectado os trilhos de tamanhos variados, onde as células serão liberadas (ainda será definido qual será o tipo de contorno aperiódico das paredes dos trilhos; provavelmente o tipo de contorno fará parte do espaço de parâmetros). Esperamos identificar, a partir de uma análise visual, para quais conjuntos de parâmetros obtêm-se os movimentos emergentes, definindo-se assim qual será o espaço de parâmetros.
Uma vez tendo uma boa intuição sobre a dinâmica do sistema partiremos para a otimização do programa, onde será utilizado o método das caixas . Nesse momento é importante já estarem definidas quais serão as grandezas a serem medidas, pois é durante a construção do programa otimizado que irá se definir como serão feitas tais medidas (o programa otimizado deve ser construído de modo que diminua o custo das medições, afim de que a otimização seja boa). Tendo o programa otimizado, iremos redefinir o espaço de parâmetros, caso achemos que seja necessário (o programa otimizado abre a possibilidade de obter mais intuição sobre o sistema, com baixo custo).
Finalmente, tendo o programa otimizado (e os métodos de medição definidos), iremos variar os parâmetros, medir para quais conjuntos de parâmetros emergem os comportamentos coletivos distintos (as transições de fases) e assim construir (mapear) o espaço de parâmetros.
Vale salientar que a metodologia detalhada acima depende de conseguirmos, utlizando o modelo proposto, obter um conjunto de parâmetros que apresente os movimentos emergentes descritos. Estudar a possibilidade do modelo apresentado não ser suficiente para descrever esses comportamentos é parte do objetivo do trabalho (vide próxima seção), assim reformulações do modelo e da metodologia do projeto podem ser esperadas.
Objetivo
Concluindo, o objetivo deste projeto se resume a duas perguntas: será possível reproduzir os comportamentos coletivos macroscópicos emergentes caterpillar like nos trilhos finos e de vórtices nos trilhos largos utilizando o modelo mínimo de Vicsek-Grégoire? Caso a resposta seja sim, quais seriam as variações no conjunto de parâmetros que possibilitariam tais transições de fase?
Esperamos que a(s) resposta(s) a essa segunda pergunta nos possibilitem construir um modelo fenomenológico para a migração de células sujeitas a restrições geométricas capaz de reproduzir comportamentos in vivo e in vitro e até de prever certos comportamentos, possibilitando assim (mais) um interplay entre estas duas áreas da ciência: a física e a biologia.
Cronograma de execução
\* Definir, in grosgrain, para quais parâmetros emergem: os movimento caterpillar like nos trilhos finos e os vórtices nos trilhos largos.
\** Definir, in finegrain, para quais parâmetros emergem: os movimento caterpillar like nos trilhos finos e os vórtices nos trilhos largos.
Bibliografia
[1] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. {\bf 75}, 1226 (1995).
[2] Reynolds, C. W. (1987) Flocks, Herds, and Schools: A Distributed Behavioral Model, in Computer Graphics, 21(4) (SIGGRAPH '87 Conference Proceedings) pages 25-34.
[3] G. Grégoire, H. Chaté, and Y. Tu, Physica (Amsterdam) {\bf181}D, 157 (2003).
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