¡We have received a new analysis of Adam Hewison on title subject:!
On Wednesday, after much hype and drama, Steve Jobs walked onstage and unveiled Apple's latest creation - the iPad. Having watched almost every key address for Apple for many years I, like many others, were disappointed that the product didn't live up to the hype. Nonetheless, Apple will sell a boatload of these products, but not as many as the iPhone.
Upon reflection, it occurred to me that Steve Jobs is changing the whole business model of Apple and I don't believe anyone has caught on to this yet.
In all the reports I've read after the launch of the iPad, I think every writer /analyst missed this key point: Steve Jobs wants to be like King Gillette.
If you don't know who King Gillette was, you may not old enough to shave. King Gillette started his business at the beginning of the century. His business model is what I believe Apple's business model will be in the future.
Long ago, King Gillette decided to practically give the razor away at or below cost, but sell the razor blades separately.
So here's what I think, I think Apple wants to give the iPhone and the iPad to as many people as possible at cost or with a small profit. Remember now, AT&T subsidized the iPhone and Apple gets a slice of the pie from every AT&T customer that has an iPhone. Now why would they do that you might ask?
The key reason, I would argue, is that Apple wants the magic of recurring revenues. This is the dream of many companies - to have millions of folks paying a small amount of money every month for using a service. What makes Apple stand out is the fact that they have an army of developers who are writing code for some very cool apps. Yes, there is an app for that. In fact, there is an app for almost every idea ever thought of.
Not only has the app store been widely successful, but Apple also has iTunes, and iBooks along with iTV coming down the road. So this is what I believe Apple's business model is going to be: with 125 million people who have giving Apple their contact and credit card information, Apple has a huge base of customers much like the newspapers and magazines did in the '60s and '70s, but on a much smaller scale. Now Apple can upsell products to those customers at will. The genius part about all of this is the fact that other people are creating products to be sold through the Apple store. Apple just reinvented the King Gillette model in a thoroughly modern way. Hat's off to you Steve.
That's my take on Apple's stealth business model.
Now let's take a look at the stock.
In my short video, I explain to you some key factors I'm watching that I think will make the difference in this market. If you have a few minutes, please take the time to watch this juggernaut of a stock and what I think is ahead for the market in the next 2 months.
As always videos are free to watch and there is no registration required.
The only request that we make is that if you find the video interesting or even disagree with the analysis, please comment on a blog. We would love to hear from you.
'Panic Boomers' Pressured to Save Now—But How?
Today reports from BP, Archer Daniels Midland, UPS and more.
President Obama unveils proposed budget for 2011 fiscal year.
The government is growing faster that we can afford, Sen. Judd Gregg (R-NH) told CNBC.
Toyota announced a fix for about 2.4 million of its 8 most popular models involved in a safety recall and sales and production suspension. Jim Lentz, president of Toyota U.S.A., discusses the fix with CNBC's Phil LeBeau.
HSBC is happy with its investment in Bank of Communications and it will remain HSBC's core investment in China, says Michael Geoghegan, group chief executive at HSBC. He outlines the bank's expansion plans, with CNBC's Amanda Drury, Maura Fogarty & Emily Chan.
surface tension that could cause it to stick.
in order to obtain their positions or territories.
pass up such offers -- even if they should know better.
said about the bankers too.
February 1, Chicago - National Futures Association (NFA) has ordered NetBlack Capital LLC (NetBlack) to withdraw from NFA membership and not reapply for NFA membership for a period of five years. NetBlack is a commodity trading advisor located in Las Vegas, Nevada. The firm must also pay a fine of $15,000 in the event it reapplies for NFA membership after five years. NFA also levied a $15,000 fine against John F. Netto, a principal and sole associated person of NetBlack.
The Decision, issued by an NFA Hearing Panel, is based on an NFA Complaint filed in August 2009 and a settlement offer submitted by NetBlack and Netto.
The Panel found that NetBlack and Netto conducted business with Jason Black (aka Jes Black), who was suspended from NFA membership as a result of a prior NFA Business Conduct Committee disciplinary action issued in February 2008. The Panel also found that NetBlack and Netto failed to maintain support for its advertised monthly rates of return as reflected in NetBlack's promotional material.
Along with the $15,000 fine, Netto is prohibited from doing business with Black for a period of thirty months for any and all firms for which Netto is principal. Additionally, for a period of twenty-four months, Netto agreed to submit all promotional material which contains performance information to NFA for review and approval prior to use.
Don't succumb to the pessimism of disbelief.
The second year after a bear market is a fine time to own stocks.
This should be a great time to own stocks.
But we forget long-term lessons and think about the recent past.
Fisher thinks 2010 will be another good year for stocks.
Here are five of his best bets.
big bear markets and recessions.
Weekend Top Story: Is a big potash play in the offing?


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How to Use Safari on an iPhone. This wikiHow teaches you how to use Apple's native web browser on your iPhone. Open the Safari app. It's a white app with the image of a blue compass.
When surfing the web on your iPhone�s Safari browser, it is common to find websites that offer content in PDF format. The Safari browser for iOS can open PDF documents but what if you need to download and save the PDF documents to read later?
22/08/2015�� The PDF documents opened from other apps (e.g. Mail, Safari) via the "Open in" feature should now appear in the "Local" and "Recent" sections. Please give it a try and let us know if you find any other issues with iOS 9 Beta.
Apple's Safari is the default web browser on iOS, so it's many people's go-to app for surfing the mobile web. Here's how to use it and customize your settings on your iPhone or iPad.
It has full Web access through an iPhone version of Safari. You can also work with email, even email with attachments, in much the same way as you can with your Mac, with the help of applications you can purchase from the Apps Store.


text:
Q: Show source code next to contentEditable div Right now, I'm trying to create a WYSIWYG HTML editor, and I need to display the source code of a contentEditable div next to the div that is being edited. I'm trying to automatically synchronize these two elements, and I'm not yet sure what the best approach would be.

test 1

<input type = "text" id = "showSourceCodeHere"></input>

Whenever the contentEditable div is updated, I want the displayed source code to be automatically updated as well, and vice-versa. Is it possible to synchronize the contents of a contenteditable div with a text box that shows the source code of the contenteditable div?

A: Here is one working solution: http://jsfiddle.net/sDdN3/14/
HTML:

test 1

<input type = "text" id = "showSourceCodeHere"></input>

JavaScript:
function test(){
    var divs = document.getElementById('showSourceCodeForThis');
        divs.onkeyup=function(event){
            document.getElementById('showSourceCodeHere').value = document.getElementById('showSourceCodeForThis').innerHTML;
        }
}

function test2(){
    var divs = document.getElementById('showSourceCodeHere');
        divs.onkeyup=function(event){
            document.getElementById('showSourceCodeForThis').innerHTML = document.getElementById('showSourceCodeHere').value;
        }
}

test();
test2();

redpajama_set_name: RedPajamaStackExchange


These first two websites are comprehensive search engines designed for students who are interested in learning more about Criminal Justice programs and careers around the country.
County Office.org It has a great number of local county and state offices with phone numbers, addresses and directions nationwide .
ASVAB practice test website The Armed Services Vocational Aptitude Battery (ASVAB) test is a required test administered to anyone interested in enlisting in any of the US Armed Service branches.


We offer a supportive, organised and creative work environment. Our developers get to work on important projects from within a couple weeks of joining and everyone on the team is encouraged to take part in continuous professional development. So whether you've been coding for twelve years or twelve weeks, what we're looking for in our team above all else is effort and passion poured into our projects.
Working with us is a great opportunity to gain broad experience in an exciting industry. There's a relatively flat hierarchy, and everyone is encouraged to contribute their own ideas as to how we can become even better at what we do.
We typically work with entrepreneurs and SMEs developing early-stage digital products with an MVP mindset, and we continue to support and iterate on their products as they grow. Since our clients vary greatly in terms of industry and the products they are seeking to develop, every day is different at Bad Dinosaur, and we get to work on some very innovative and exciting projects.
Our headquarters is located at 6/3 Beaverhall Road, in the lovely Canonmills area of Edinburgh. You can get to us on Lothian bus numbers 8, 23, 27 or 36 from the city centre and we're also located close to several main cycle routes.
Our office is equipped with a fully stocked kitchen, espresso machine, games consoles, VR, bean bag chairs and plenty of whiteboards. On Fridays we treat our team to lunch from a local restaurant have social drinks after work. We also get weekly visits from Shoshana the cockapoo to keep up our spirits.
If you want to get out of the office for a bit or need to meet a client elsewhere, we also have an office at Clockwise in Glasgow, which has nice meeting spaces and great coffee.
To see our vacancies, keep an eye on our Indeed page and our social media accounts.


text:
Home / Press Releases / Israeli demolition campaign in three towns across the West Bank
Israeli demolition campaign in three towns across the West Bank
in Press Releases April 14, 2010
Popular Struggle Coordination Committee
Ali Salim Mousa stands in the middle of the rubble of his house, demolished on 14.04.2010, in Al Khader.
In an aggravation of Israeli policy of home demolitions, a house and two shops were razed in the Central West Bank village of Hares this morning. An additional house was reduced to rubble in the village of alKhadder, West of Bethlehem and a 1,000 square meters factory was demolished in the town Beit Sahour.
In what seems as a coordinated move, Israeli forces carried out demolitions in two different areas of the West Bank today, rendering at least 16 people homeless. In recent months, international pressure has cause a significant decline in the demolition of Palestinian houses in the Israeli-controlled Area C of the West Bank. Israel uses its statutory authority in Area C, which spans over 60% of the West Bank, to dramatically limit Palestinian development. Palestinians fear that today's consorted demolitions may be the opening salvo in a provocative change in Israeli policy.
Mahmoud Zwahare a popular committee member from the Bethlehem region said during the demolitions that "Israel keeps claiming it strives for peace and constantly complains about Palestinian incitement and violence. It is doing so while carrying destructive and irreversible steps on the ground against ordinary civilians. The demolitions today has nothing to do with the security of Israelis and everything to do with provocation and injustice".
A convoy of eight armored military jeeps and a D9 bulldozer entered the village of Hares in the early morning today and advanced towards the newly built house of Maher Sultan. The house, which Sultan had just finished constructing after five years, was to be home for himself, his wife and their five children. The two story house was quickly demolished by the bulldozer, which left nothing but rubble behind it.
The demolition order was posted on Sultan's house a month ago, citing a Mohammed Mansour as the owner of the house, which complicated to procedures to stop the demolition. At the time of the demonstration, Sultan was actually at the DCO in Tulkarem to try an negotiate an injunction, unaware that his home is being razed.
After completing the demolition of Sultan's house, the Israeli forces continued to demolish two stores in the outskirts of the village.
Almost simultaneously, a massive contingent of Israeli forces invaded the town of alKhadder, West of Bethlehem. The massive Israeli bulldozer demolished the house of Ali Mousa, which was home to nine people, including a one year old baby, as soldiers prevented anyone from nearing the house – including the family's lawyer, who showed soldiers a 2006 court-issued injunction on the demolition.
After completing the demolition, an Israeli Civil Administration officer who was present at the scene informed people that more house demolitions will be carried in the near future.
Shortly after the alKhadder demolition, forces lead by the Israeli Civil Administration demolished a factory in the town of Beit Sahour. Roughly a year ago, Omar Ayyoub, the owner of the factory was served a halt-construction order by the civil administration, which he complied with and have been fighting ever since. When the bulldozers arrived today he pleaded with the officer in charge to stop the demolition, or at least present him with a valid demolition order. The officer refused and ordered to remove Ayyoub from the scene.
Over 60 percent of the West Bank is currently classified as Area C, in which, under the Oslo accords, Israel has complete control, over both civil and security issues. According to the UN Office for the Coordination of Humanitarian Affairs (OCHA) some 70 percent of Area C, or approximately 44 percent of the West Bank, has been largely designated for the use of Israeli settlements or the Israeli military. The Israeli authorities generally allow Palestinian construction only within the boundaries of an Israeli-approved plan and these cover less than one percent of Area C, much of which is already built-up. As a result, Palestinians are left with no choice but to build "illegally" and risk demolition of their structures and displacement.
According to information released by the Israeli State Attorney's Office in early December 2009, approximately 2,450 Palestinian-owned structures in Area C have been demolished due to lack of permit over the course of the past 12 years.
Ethnic Cleansing House Demolition	2010-04-14
Tagged with: Ethnic Cleansing House Demolition
Israeli apartheid in Masafer Yatta: raids, demolitions, arrests and beatings
Statement on "postponement" of Regavim London talk due to activist pressure
Ten people arrested from Beit Kahel: the village targeted with violent night raids and demolition threats

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Wondering how to write an essay? We have collected essay writing tips for different essay types and different essay topics. Throughout your academic career, you will usually be asked to write an essay. You may work on an assigned essay for class, enter an essay contest or write essays for college admissions. This section explores different essay types and will show how to write an essay on different topics.
An essay is usually an extended piece of written work (undergraduate essays vary in length from 1000 to 5000 words), and is supposed to involve analysis, clear argument, and research. In the essay you bring these elements together. You are supposed to respond to your question or task by researching as widely as time and resources allow, in order to be able to come up with evidence, and perhaps different viewpoints, which you then analyze or subject to critical thought. An essay is not just about providing a 'right answer', or demonstrating what you know. In most cases what is being assessed is the process – of researching, analyzing and evaluating your material, and producing an argument which reflects both the evidence provided and your ability to think critically about it.
We'll look at each type of paragraph, and at transitions, the glue that holds them together. Learn How to Write a 5-Paragraph Essay.
The argumentative essay is a genre of writing that requires the student to investigate a topic; collect, generate, and evaluate evidence; and establish a position on the topic in a concise manner. Argumentative essays allow writers to express their opinion on a topic and support that opinion with strong logic and evidence. Read on to discover the outline for how to write an argumentative essay, and see examples of how to construct each part of this essay. Learn How to Write an Argumentative Essay.
Above all, a review makes an argument. The most important element of a review is that it is a commentary, not merely a summary. It allows you to enter into dialogue and discussion with the work's creator and with other audiences. You can offer agreement or disagreement and identify where you find the work exemplary or deficient in its knowledge, judgments, or organization. You should clearly state your opinion of the work in question, and that statement will probably resemble other types of academic writing, with a thesis statement, supporting body paragraphs, and a conclusion. Learn How to Write a Book Review.
A classification or division essay groups objects, people, or events by the characteristics that they share. Usually, the writer begins with a group of people or things and then separates them into subgroups or types. There are several steps to writing an effective division or classification essay.
Learn How to Write a Classification Essay.
The purpose of a descriptive essay is to create an impression on the reader. This type of theme should excite the reader's senses (touch, taste, sight, sound, and smell) through the use of vivid words and details.
Good description is essential for all writing, from business reports to personal letters to poems to persuasive essays. Description commits writers to honesty by pinning down vague statements with concrete details. Honest description shows the world as it is with all its warts, pimples, and secret beauties. It challenges safe stereotypes and preconceptions by getting down on its knees to scrutinize the topic closer, to find and report what's really there. In persuasive or research papers, powerful description can support a thesis in combination with statistics and other facts. Learn How to Write a Descriptive Essay.
Expository writing is a type of writing that is used to explain, describe, give information, or inform. The text is organized around one topic and developed according to a pattern or combination of patterns. The writer of an expository text cannot assume that the reader or listener has prior knowledge or prior understanding of the topic that is being discussed. Since clarity requires strong organization, one of the most important mechanisms to improve skills in exposition is to improve the organization of the text.
The purpose of the expository essay is to explain a topic in a logical and straightforward manner. Without bells and whistles, expository essays present a fair and balanced analysis of a subject based on facts—with no references to the writer's opinions or emotions. Learn How to Write an Expository Essay.
An informative essay is a form of writing that teaches the reader about a topic in an unbiased manner. Typically, this type of essay will include an introduction, a few body paragraphs, and a conclusion. The body of the essay generally will contain facts that are well-researched and come from reliable sources.
There are many uses for an informative essay. For example, it may be used to inform readers about a product, a process, a person, or an event. Although it may be used to discuss a controversial issue, it may not be used to express the writer's opinion about that issue. In those cases, the writer should present both sides of the issue in an unbiased manner — the reader should not be able to infer the writer's opinion on the topic after reading an informative essay. Learn How to Write an Informative Essay.
A narrative essay is an essay that tells a story about a specific event or experience. Narratives have a point, and the narrative (story) is used to convey the point. A narrative includes all the key events of the story, presented in time order. The narrative essay is more than just a listing of events; it often uses descriptive and sensory information to make the narrator's point and to make the story real for the reader. Consequently, narratives are often subjective rather than objective.
A narrative essay is a format in which the author tells, or narrates, a story. They are non-fictional and deal with the author's personal development. Unlike other forms of writing, using the first person is acceptable in narrative essays. Learn How to Write a Narrative Essay.
Persuasive essay writing has a bad reputation. At its worst we picture a slimy hoodlum twisting someone's arm to "persuade" him to tell where the diamonds are. We think of battleships and bombers hovering off the coast of a tiny country to "persuade" its government to change its policies. Cigarette advertisements in magazines "persuade" men that one brand will make them feel as if they're riding a snorting stallion on the range while another brand will make women feel daring and rebellious. How about empty political campaign speeches or a father screaming at his late, tipsy teenager at 2 A.M.? Persuasion is commonly pictured as forcing, tricking, seducing, or lecturing people to buy or do something not really in their best interests. Too often these tactics work. But that's not the kind of persuasion we're interested in here.
A persuasive essay is an essay used to convince a reader about a particular idea or focus, usually one that you believe in. When writing a persuasive essay, a writer takes a position FOR or AGAINST an issue and writes to convince the reader to believe or do something. Learn How to Write a Persuasive Essay.
Let's be honest: most students hate research papers. They think it an assignment that is both picky and tedious. Wrestling with the correct format of a footnote drives them loopy. That, in a nutshell, is how you really feel about the research paper. Many students feel exactly the same. Yet for all the anxiety that the research paper provokes, it has outlasted generations of its student haters. Obviously it must be good for something, or else it would have been swept away long, long ago.
In fact, the research paper is an excellent tool for learning about a topic of your choice. Writing it will expose you to the rigors of research, acquaint you with the protocol of making correct citations to sources consulted, and teach you how to forge a mishmash of researched opinions into a single, coherent viewpoint. Of course, it is possible that your instructor will assign a specific topic for your paper, but typically topic choice is left up to the writer. Finding and shaping the final topic is usually regarded as a test of the student's judgment. The student who chooses a vast topic, such as wars throughout the ages, has taken on too big a job. On the other hand, the student who chooses to write on the history of the tire iron is proposing a topic that is too small. So how to write a research paper? Our writing guide is arranged following the logical steps of writing process. Learn How to Write a Research Paper.
The essay is probably the commonest form of assessment at universities. It is now used more than exams, and it is not specific to any discipline area: you will be writing essays if you are doing subjects in anthropology, communication, economics, business, health science, tourism, engineering or education. Essays are such a popular form of assessment because they are useful for determining how well you have understood – not just memorized. Whereas exams of the short-answer kind assess the ability to learn facts, essays allow you to demonstrate important literacies, such as being able to research, select evidence and use examples, and evaluate different points of view. In essays, you demonstrate your ability to use the discourses of the discipline. These kinds of literacies are those which university students are supposed to have, and employers want. The following guide will show how to write an essay on selected fields of study.
Anthropology is the study of human groups and cultures, both past and present. Anthropology shares this focus on the study of human groups with other social science disciplines like political science, sociology, and economics. What makes anthropology unique is its commitment to examining claims about human 'nature' using a four-field approach. The four major subfields within anthropology are linguistic anthropology, socio-cultural anthropology (sometimes called ethnology), archaeology, and physical anthropology. Each of these subfields takes a different approach to the study of humans; together, they provide a holistic view. So, for example, physical anthropologists are interested in humans as an evolving biological species. Linguistic anthropologists are concerned with the physical and historical development of human language, as well as contemporary issues related to culture and language. Archaeologists examine human cultures of the past through systematic examinations of artifactual evidence. And cultural anthropologists study contemporary human groups or cultures. Learn How to Write an Anthropology Essay.
Courses in communication studies combine material from the humanities, fine arts, and social sciences in order to explain how and why people interact in the ways that they do.
Learn How to Write a Communication Essay.
History is everything that happened in the past: dates, facts, timelines, and the names of kings, queens, generals, and villains. For many students, the word "history" conjures up images of thick textbooks, long lectures, and even longer nights spent memorizing morsels of historical knowledge.
For your instructors in the history department, however, history is a fascinating puzzle with both personal and cultural significance. The past informs our lives, ideas, and expectations. Before shrugging off this abstract notion, ask yourself another "easy" question: Why are you here at your college or university?
Maybe you're at your college because it was the best school that accepted you, or because it has great sports teams. In the big picture, however, you are here because of history, i.e., because of past events and developments. You are here (on the planet) because two people's lives collided—in the past. You may be here (in America) because you or some ancestor crossed an ocean several weeks, years, decades or centuries ago. You are here (in college or university) because, two hundred years ago, some people pooled their ideas, energy, and money to dig a well, collect some books, and hire some professors. You are here (at an institution of higher education) because long ago, some German scholars laid the groundwork for what we call the "modern university." Learn How to Write a History Essay.
Writing an analysis of a piece of literature can be a mystifying process. First, literary analyses (or papers that offer an interpretation of a story) rely on the assumption that stories must mean something. How does a story mean something? Isn't a story just an arrangement of characters and events? And if the author wanted to convey a meaning, wouldn't he or she be much better off writing an essay just telling us what he or she meant?
But if the meanings were always as clear as they are in parables, who would really need to write a paper analyzing them? Interpretations of fiction would not be interesting if the meanings of the stories were clear to everyone who reads them. Thankfully (or perhaps regrettably, depending on your perspective) the stories we're asked to interpret in our classes are a good bit more complicated than most parables. They use characters, settings, and actions to illustrate issues that have no easy resolution. They show different sides of a problem, and they can raise new questions. In short, the stories we read in class have meanings that are arguable and complicated, and it's our job to sort them out. Learn How to Write a Literature Essay.
Learn How to Write a Philosophy Essay.
Learn How to Write a Political Science Essay.
Writing in psychology is similar to other forms of scientific writing in that organization, clarity, and concision are important. The Psychology Departments at colleges and universities have a strong research emphasis, so many of your assignments will focus on synthesizing and critically evaluating research, connecting your course material with current research literature, and designing and carrying out your own studies. Learn How to Write a Psychology Essay.
Before you can write a clear and coherent sociology paper, you need a firm understanding of the assumptions and expectations of the discipline. You need to know your audience, the way they view the world and how they order and evaluate information. So, without further ado, let's figure out just what sociology is, and how one goes about writing it.
Unlike many of the other subjects at colleges and universities, such as history or English, sociology is a new subject for many students. Therefore, it may be helpful to give a quick introduction to what sociologists do. Sociologists are interested in all sorts of topics. For example, some sociologists focus on the family, addressing issues such as marriage, divorce, child-rearing, and domestic abuse, the ways these things are defined in different cultures and times, and their effect on both individuals and institutions. Others examine larger social organizations such as businesses and governments, looking at their structure and hierarchies. Still others focus on social movements and political protest, such as the American civil rights movement. Finally, sociologists may look at divisions and inequality within society, examining phenomena such as race, gender, and class, and their effect on people's choices and opportunities. As you can see, sociologists study just about everything. Thus, it is not the subject matter that makes a paper sociological, but rather the perspective used in writing it. Learn How to Write a Sociology Essay.
Students may not distinguish between the comments and corrections referring to relatively minor errors of style and language use an comments on more fundamental aspects, such as the quality of the argument, structure, and depth of understanding.
Students are mainly interested in the mark awarded. They may not take much notice of the comments, nor make use of them to improve their next assignment.
Introduction. This prepares your reader for what is t o come. It orients your argument t o the set topic (e.g. by explaining what you understand by the question).
Main body. This is where you bring in the evidence o'f your research, and put together your argument's.
Conclusion. This brings the essay t o an end. The reader is reminded in some way of every major point you made or issue you considered.
Bibliography. Any piece of work that contributed to your essay should be included. This does not include only books – films, magazines, and newspapers should be included if you made use of them, even if you don't quote them directly.
You will need to find out what requirements there are about format and style in the particular discipline (or unit). For instance, you will probably be told that you must have a separate title page and use a certain referencing convention (e.g. the Harvard system). Some disciplines, notably the humanities, tend to disapprove of such devices as subheadings or lists using numbers or bullet points.
Note that oral presentations, too, should have a distinct introduction, body and conclusion (in your mind, even if the transitions between them are not noticeable to the audience). Oral presentations do not have bibliographies, but you should not use other people's ideas-or provide contestable facts and figures-without briefly indicating their source.


We propose a more realistic approach to trip planning for tourist applications by adding category information to points of interest (POIs). This makes it easier for tourists to formulate their preferences by stating constraints on categories rather than individual POIs. However, solving this problem is not just a matter of extending existing algorithms. In our approach we exploit the fact that POIs are usually not evenly distributed but tend to appear in clusters. We develop a group of efficient algorithms based on clustering with guaranteed theoretical bounds. We also evaluate our algorithms experimentally, using real-world data sets, showing that in practice the results are better than the theoretical guarantees and very close to the optimal solution. Copyright 2014 ACM.


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Home > Rides > Cars
This $1.4-Million Ute Could Become Australia's Most Expensive
The ute ceased to meet a $1.25-Million reserve at auction.
Ben McKimm, 18 Mar 2022 | News
You should've seen our face when we stumbled across this timeless beast online. Then you should've seen our face after we saw the $1.4-Million price. And THEN you should've seen our face when we realised this is the same ute causing an auction room ruckus at Lloyds only a couple of months ago. AND THEN you should've seen our face when we realised the seller turned down a $1.25-Million dollar offer on the car after it didn't meet reserve! Introducing the HSV destined to become Australia's most expensive ute, ever.
Related: This Honda Civic recently sold for $105,000 and we have no idea what's going on?
Image: Lloyds
It was in February 2021 that we last saw a Holden Special Vehicles GTS R W1 ute go across the auctions at Lloyds. A significant occasion that saw an HSV GTS R W1 ute break the record and claimed the title of 'Australia's Most Expensive Ute' when it rolled off the auction block for a staggering $1.05-Million, but those close to the auction knew this record wouldn't stand for long with a lineup of Aussie icons set to hit the stands only a few months later.
Fast forward to August 2021 and a lineup of rare HSVs were all set ready to break auction records. One particular model stood out from the pack, a special ute with black wheels, a 'Yellah' paint scheme that harks back to the famous HSV's of yesteryear, manual transmission, 474KW Supercharged LS9, and dealer delivery KM's. This 2017 Holden Special Vehicles GTS R W1 in manual is as rare as they come, build #001 and the only W1 Maloo Ute to be painted in-house at HSV in the famous VS GTSR XU3 'Yellah' before being released. Quite frankly, it doesn't get more Aussie (or rarer) than this so we expected a big price tag and a new 'Most Expensive Ute' king. And we got it.
Related: Only a couple of months after releasing, people are asking crazy money for Nissan GT-R 'T-Specs'.
Reaching a hammer price on the day of $1,250,000 AUD, surprisingly, it never met the reserve. According to Lloyds, negotiations continued over the following days, but a price was clearly never agreed upon. Begging the question, if you got offered $1,250,000 AUD for your 2017 HSV GTS R W1 would you take it? Our answer would be a resounding yes, but owner and seller of this very same ute doesn't think so, posting the car up for sale privately with the current carsales.com.au ad reading;
"THIS CAR NEEDS NO INTRODUCTION…..!!
The worlds rarest HSV.
This was published that it sold, but the ute never met my reserve at Auction. I declined to sell after the auction due to a private bidding war post-auction which was too much muck around.
This car deserves more respect than that so I said not for sale anymore…!!!"
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text:
\subsection*{Protrusion dynamics drive confined cell migration}

To investigate the dynamics of cell shapes, protrusions, and nucleus in confined migration, we study the migration dynamics of single MDA-MB-231 breast carcinoma cells confined to two-state micropatterns (Fig.~\ref{fig1}a). These patterns consist of two adhesive islands connected by a thin adhesive bridge, allowing us to study how migrating cells respond to constrictions in the extra-cellular environment. We use time-lapse phase-contrast microscopy and fluorescent staining of the cell nuclei to investigate the joint dynamics of cell shape and nucleus motion. We find that the motion of the nucleus is correlated with the growth of a protrusion across the constriction of the pattern, suggesting that the protrusion dynamics of these cells are key to understanding cell migration dynamics.

To quantify these protrusive dynamics, we first isolate cell shapes from bright-field microscopy image stacks using a convolutional neural network with a U-Net architecture~\cite{Ronneberger2015} (Methods). This segmentation procedure allows us to accurately determine the 2D shape of the cells as a function of time (Fig.~\ref{fig1}a). To identify protrusions, we classify those components of the cell shape added in each time step as protrusive areas (Fig.~\ref{fig1}b,c, Methods, Supplementary Movie S3) \cite{Machacek2006}. During the traversal of cells across the constriction, large protrusive areas are formed at the leading edge of the cell. Importantly, due to the micropattern geometry, most protrusive activity is in the $x$-direction along the long axis of the pattern (Appendix~\ref{sec_image}). Thus, to provide a low-dimensional representation of the protrusion dynamics, we define the effective protrusion position $x_\mathrm{p}(t)$ as the $x$-component of the geometric center of protrusive area (Fig.~\ref{fig1}c), referred to as the protrusion from here on. Indeed, the protrusion trajectories serve as an indicator of the protrusive dynamics of the cells, as shown by an overlay with the kymograph of the microscopy images (Fig.~\ref{fig1}d, Supplementary Movie S3). In addition, we track the trajectories of the cell nucleus. While the cells also perform retractions at the trailing edge, we find that these are strongly correlated with the motion of the nucleus with near-zero time-lag, and therefore do not contain significant additional information (Fig.~\ref{fig1}g, Appendix~\ref{sec_image}). Thus, we  restrict our analysis to the nucleus-protrusion dynamics. This analysis pipeline gives access to a large data set of low-dimensional trajectories of cell nucleus and protrusion dynamics (1400 trajectory pairs with duration up to 50h), allowing an in-depth statistical analysis of the cellular dynamics.

The joint nucleus and protrusion trajectories reveal that these cells tend to migrate across the constriction in a stereotypical manner: first, the protrusion grows slowly across the constriction, after which the nucleus rapidly follows (Fig.~\ref{fig1}e). The nucleus motion exhibits weaker fluctuations than the protrusions and responds to the protrusions with a time delay, as quantified by the cross-correlation function $\langle x_\mathrm{n}(t) x_\mathrm{p}(t+T) \rangle$ (Fig.~\ref{fig1}g). The stereotypical migration pattern is reflected as a ring-like structure in the joint probability distribution of nucleus and protrusion positions $p(x_\mathrm{n},x_\mathrm{p})$  (Fig.~\ref{fig1}f). While the most likely states are where both nucleus and protrusion occupy one island, there is significant probability along the path where the protrusion first crosses the constriction and reaches the other island, followed by the traversal of the nucleus. In contrast, there is  low probability of observing both protrusion and nucleus in the constriction. Together, these results indicate that the confined migration dynamics exhibit a stereotypical `protrusion-nucleus cycling' represented as paths in $x_\mathrm{n}x_\mathrm{p}$-space.

\begin{figure}[ht]
	\includegraphics[width=0.5\textwidth]{fig3}
	\centering
		\caption{
		\textbf{Nucleus velocity maps constrain model candidates.}
		\textbf{a.} Schematic of the model. Arrows indicate the forces acting on the cell at positions $x_\mathrm{n}$ and $x_\mathrm{p}$.
		\textbf{b.} Experimental nucleus velocity map (NVM), calculated as the conditional average of the nucleus velocity as a function of nucleus and protrusion positions, $\langle \dot{x}_\mathrm{n} | x_\mathrm{n},x_\mathrm{p} \rangle$, shown with interpolation.
		\textbf{c.} Double-well potential $W(x_\mathrm{n})=Q(1-(x_\mathrm{n}/x_0)^2)^2$, where $Q$ determines the height of the energy barrier, and $x_0$ the positions of the minima. Image indicates the dimensions of the micropattern, and shows a fluorescence microscopy image of the actin cytoskeleton of a confined cell (LifeAct-GFP-transfection).
		\textbf{d.} NVM predicted by the energy potential. Parameters are determined by a best fit to the experimental NVM with two free fit parameters $k/\zeta_n$ and $Q$ (Appendix~\ref{sec_nucleus}).
		\textbf{e.} Cuts of the NVM along the horizontal lines indicated in panel \textbf{d}, showing $\dot{x}_\mathrm{p}$ as a function of $x_\mathrm{n}$ for different $x_\mathrm{p}$. Dots: Experiment, solid lines: deformation model prediction.
		\textbf{f.} Spatially variable friction $\gamma(x_\mathrm{n}) = \gamma_\mathrm{min} + \frac{1}{2}(1-\gamma_\mathrm{min}) \left(1 - \cos \left(x_\mathrm{n} \pi / L_\mathrm{system} \right) \right)$ used in the adhesion landscape model, where $\gamma_\mathrm{min}$ is the friction at the center of the constriction, and $L_\mathrm{system}$ is the total length of the micropattern.
		\textbf{g.} NVM predicted by the adhesion landscape model with two free fit parameters $k/\zeta_n$ and $\gamma_\mathrm{min}$.
		\textbf{h.} Same plot as in panel \textbf{e} for the adhesion landscape model.
						 }
	\label{fig2}
\end{figure}

\subsection*{Confined cells migrate in an adhesion landscape}

We aim to develop a mechanistic theory to describe how the coupled stochastic dynamics of cell nucleus and protrusion determine the confined migration of cells. Our strategy will be to postulate simple model candidates based on known cellular processes, physical principles, and symmetry arguments, which we systematically and quantitatively validate with experimental data. This approach allows us to rule out a whole range of possible alternative models, and identify a promising mechanistic model with predictive power for the protrusion-nucleus dynamics of confined cells.

Experimentally, we find that fluctuations in the nucleus velocities $\dot{x}_\mathrm{n}$ are small compared to the average components. By comparison, the fluctuations in the protrusion velocities $\dot{x}_\mathrm{p}$ are much larger and dominate over deterministic contributions to the protrusion velocities (Appendix~\ref{sec_whitenoise}). Thus, we consider a model in which the intrinsic stochasticity of the system stems from the polarity dynamics driving the protrusion. We expect forces on the nucleus to arise due to two main contributions: coupling to the cell protrusion~\cite{Crisp2006,Caswell2018,Davidson2020,Davidson2021}, and the effect of the confining micropattern. Similarly, protrusions couple to the cell nucleus~\cite{Caswell2018}, and may be sensitive to the external environment. Taken together, considering force balance at $x_\mathrm{n}$ and $x_\mathrm{p}$ (Fig.~\ref{fig2}a), we obtain
\begin{align}
\label{eqn_eom_n}
\zeta_\mathrm{n} \dot{x}_\mathrm{n} &= f_\mathrm{c}(x_\mathrm{n},x_\mathrm{p}) + f_\mathrm{n}(x_\mathrm{n})  \\
\label{eqn_eom_p}
\zeta_\mathrm{p} \dot{x}_\mathrm{p} &= -f_\mathrm{c}(x_\mathrm{n},x_\mathrm{p}) + f_\mathrm{p}(x_\mathrm{p}) + P(t)
\end{align}
where $\zeta_\mathrm{n}, \zeta_\mathrm{p}$ are the friction coefficients of nucleus and protrusion, respectively, $f_\mathrm{c}$ is the coupling between nucleus and protrusion, and $f_\mathrm{n,p}$ are additional forces acting on each nucleus and protrusion due to the confinement. Additionally, we assume the protrusion to be driven by a stochastic active force $P(t)$, which serves as a minimal implementation of the time-dependent forces driving protrusion formation, such as the active pushing force due to actin polymerization~\cite{Pollard2003a,CallanJones2016}. This active force determines the instantaneous direction of polarization in which protrusions are generated, and we therefore refer to it as the cell polarity.

To constrain our model step-by-step, we start with the dynamics of the cell nucleus. In migrating cells, the motion of the nucleus is coupled to the dynamics of the leading edge, for example, through material stresses in the cytoskeleton connecting protrusion and nucleus~\cite{Crisp2006,Caswell2018,Davidson2020a,Davidson2020}, or through mechanical feedback processes coupling the leading and trailing edge of the cell~\cite{Tsai2019}. As a minimal model for this coupling, we consider a linear elastic spring, similar to previous work~\cite{Ron2020a,Sens2020}.

It is less clear, however, how to incorporate the effect of the confining micropattern on the dynamics. Physically, we consider two distinct ways to couple the cell dynamics to geometry. First, a conservative force, corresponding to a double-well potential $W(x_\mathrm{n})$, with minima on the adhesive islands and a barrier around the constriction (Fig.~\ref{fig2} c), can provide a model for the contribution due to cell deformations during the transition. Such deformation dynamics of cells are frequently modelled using effective Hamiltonians including the surface and line tension of the cell~\cite{Albert2014,Bi2014,Bi2015,Segerer2015,Goychuk2018}, which would suggest that the deformed state of the cell in the constriction is associated with an increased energy. Second, the difference in adhesive area available to the cell on the island and in the constriction could lead to a dissipative force corresponding to a spatially variable friction coefficient~\cite{Tawada1991,Reboux2008}. Mesenchymal migration exhibits mature focal adhesions at the cell rear, where the nucleus typically resides~\cite{Lehnert2004a,Balaban2001}. These adhesions can only form within the micropatterned area, and we therefore expect the adhesiveness to be largest on the islands and smallest at the center of the constriction.

\begin{figure}[h]
	\includegraphics[width=0.45\textwidth]{fig4}
	\centering
		\caption{
		\textbf{Adhesion landscape model predicts dynamics with varying constriction width.}
		\textbf{a.} Friction profiles $\gamma(x_\mathrm{n})$ with increasing $\gamma_\mathrm{min}$ as a model for increasing bridge width. The value of $\gamma_\mathrm{min}$ is fitted for the narrowest bridge ($W=4 \ \si{\micro\meter}$; $\gamma_\mathrm{min} \approx 0.2, k/\zeta_n \approx 0.6$ h$^{-1}$). For the widest system without constriction, we take a flat profile and take intermediate values of $\gamma_\mathrm{min}$ for intermediate widths, such that $\gamma_\mathrm{min}$ scales linearly with the available area (Appendix~\ref{sec_nucleus}).
		\textbf{b.} Sketch of confinement geometries with increasing bridge widths $W=4,7,12,22,35 \ \si{\micro\meter}$ (from left to right); brightfield microscopy images of MDA-MB-231 cells migrating in these geometries with cell outline in pink and nucleus position in blue, and geometry in white. Scale bar: $25  \ \si{\micro\meter}$.
		\textbf{c.} Predicted and experimental nucleus velocity maps (NVM) $\langle \dot{x}_\mathrm{n} | x_\mathrm{n},x_\mathrm{p} \rangle$. Plotted with the same color axis as shown in Fig.~\ref{fig2}.
		\textbf{d.} Predicted and experimental cuts of the NVM, showing $\dot{x}_\mathrm{p}$ as a function of $x_\mathrm{n}$ for different $x_\mathrm{p}$ as described in Fig.~\ref{fig2}.
		 }
	\label{fig3}
\end{figure}

To test the energy potential contribution, we consider the equation of motion for the cell nucleus
\begin{align}
\label{eqn_doublewell}
\zeta_\mathrm{n} \dot{x}_\mathrm{n} = k(x_\mathrm{p}-x_\mathrm{n}) - \partial_{x_\mathrm{n}} W(x_\mathrm{n})
\end{align}
This equation makes a concrete prediction for how the nucleus velocity $\dot{x}_\mathrm{n}$ varies with the positions of nucleus and protrusion. To test this prediction directly on the experimental data, we determine the average velocity of the cell nucleus as a function of $x_\mathrm{n}$ and $x_\mathrm{p}$, $\langle \dot{x}_\mathrm{n} | x_\mathrm{n},x_\mathrm{p} \rangle$, which we term the \emph{nucleus velocity map} (NVM) (Fig.~\ref{fig2}b). Importantly, with this approach based purely on the nucleus velocities, we can determine the deterministic nucleus dynamics (Eq.~\eqref{eqn_eom_n}) without making  assumptions about the protrusion and polarity dynamics (Eq.~\eqref{eqn_eom_p}). However, we find that the NVM predicted by the energy potential fails to capture the experimental data, as it does not predict the characteristic acceleration of the nucleus in the constriction (Fig.~\ref{fig2}d,e). This approach similarly fails for more general non-linear elastic couplings between nucleus and protrusion (Appendix~\ref{sec_nucleus}). Therefore, we conclude that such potential energy models alone are not able to recover the cellular dynamics in this setup.

To test the possible contribution of differences in local adhesion, a simple model is a spatially variable friction coefficient:
\begin{align}
\label{eqn_adhesion}
\zeta_\mathrm{n} \gamma(x_\mathrm{n}) \dot{x}_\mathrm{n} = k(x_\mathrm{p}-x_\mathrm{n})
\end{align}
where $\gamma(x_\mathrm{n})$ ensures lower friction in the constriction (Fig.~\ref{fig2}f). This model provides an excellent fit to our data, and captures the characteristic increase in nucleus speeds during traversal (Fig.~\ref{fig2}g,h). The resulting fit parameters give a typical time-scale for the nucleus motion $\zeta_n/k \approx 1.7$h, which is in approximate agreement with known turn-over times of focal adhesions~\cite{Stricker2013a,Stehbens2014}. This time-scale is reduced in the constriction due to the reduced number of adhesions formed by the cell body around the nucleus when it is in the constriction, causing the acceleration of the nucleus during traversal. Taken together, these results indicate that a dissipative force arising from a spatially variable adhesion landscape is a key component of the effect of the confining constriction on migration dynamics, which in our setup appears to dominate over possible contributions due to cellular deformations.

\subsection*{Adhesion landscape model captures dependence of nucleus dynamics on constriction width}

The adhesion landscape model (Eq.~\eqref{eqn_adhesion}) makes a simple, intuitive prediction. As we widen the constricting bridge of the micropattern, more adhesive area becomes available, thereby reducing the variations in the friction profile (Fig.~\ref{fig3}a). In the  limit where the constriction has the same width as the islands, we expect a uniform adhesiveness profile. Accordingly, we predict the acceleration of the cell nucleus observed on thin bridges (Fig.~\ref{fig2}) to decrease with the increasing adhesiveness of a wider bridge, and to completely disappear for constant adhesiveness (Fig.~\ref{fig3}c,d). In this limiting case, we expect migration dynamics that are completely determined by the linear elastic coupling between nucleus and protrusion (last panel Fig.~\ref{fig3}d).

To challenge the predictive power of the adhesion landscape model, we perform experiments with varying bridge width (Fig.~\ref{fig3}b). Importantly, the nucleus velocity maps inferred from these experiments are well predicted by the model, and exhibit the predicted decreasing maximum nucleus speed in the constriction (Fig.~\ref{fig3}c,d). On the rectangular micropattern without constriction, we find an almost linear profile of the nucleus speed with position, as predicted theoretically. This further supports our model of a linear elastic nucleus-protrusion coupling. In summary, the adhesion landscape model has predictive power for confining geometries with varying constriction width.

\subsection*{Protrusions driven by time-correlated polarity}

Having determined how the dynamics of the nucleus couples to the confinement and protrusion, we next investigate the dynamics of the protrusion itself (Eq.~\eqref{eqn_eom_p}). As a minimal model for the protrusion dynamics, we postulate a coupling to the cell nucleus equal and opposite to the coupling introduced for the cell nucleus (Eq.~\eqref{eqn_adhesion}). In addition, we enforce a potential $V(x_\mathrm{p}) = (x/x_\mathrm{boundary})^8$ to confine the protrusion between the boundaries of the micropattern (Appendix~\ref{sec_protrusion}):
\begin{align}
\label{eqn_protrusion}
\zeta_\mathrm{p} \dot{x}_\mathrm{p} = -k(x_\mathrm{p}-x_\mathrm{n}) - \partial_{x_\mathrm{p}} V(x_\mathrm{p}) + P(t)
\end{align}
Thus, we assume that both the friction and the potential term of the protrusion are insensitive to the presence of the constriction (Supplementary Fig. S7). Importantly, however, we anticipate that the polarity $P$ may couple to the confinement, as it models the active driving of the protrusion by the migration machinery, including actin polymerization and the diffusion of polarity cues~\cite{Pollard2003a,CallanJones2016}, which may be sensitive to the geometry of the confinement.

Similar to our approach to the nucleus dynamics, this protrusion model provides a prediction for the average protrusion velocity as a function of $x_\mathrm{n}$ and $x_\mathrm{p}$, $\langle \dot{x}_\mathrm{p} | x_\mathrm{n},x_\mathrm{p} \rangle$, which we term \emph{protrusion velocity map} (PVM). According to our general model ansatz (Eq.~\eqref{eqn_eom_p}), unlike the NVM, the PVM consists of several components, including the polarity dynamics:
\begin{equation}
\label{eqn_PVM}
\langle \dot{x}_\mathrm{p} | x_\mathrm{n},x_\mathrm{p} \rangle = -f_\mathrm{c}(x_\mathrm{n},x_\mathrm{p}) + f_\mathrm{p}(x_\mathrm{p}) + \langle P | x_\mathrm{n},x_\mathrm{p} \rangle
\end{equation}
Since the polarity term $\langle P | x_\mathrm{n},x_\mathrm{p} \rangle$ does not average to zero for time-correlated polarities, we cannot in general disentangle the contributions to the protrusion dynamics based on the PVM~\cite{Lehle2018,Frishman2018}. Instead, we will constrain the polarity dynamics by systematically testing models of increasing complexity.

We first show that the data cannot be captured by the simplest possible stochastic polarity dynamics: a Gaussian white noise (WN) process $P_\mathrm{WN}= \sigma \xi(t)$ with $\langle \xi(t)\rangle=0$ and $\langle \xi(t)\xi(t') \rangle=\delta(t-t')$. In this case,  $\langle P_\mathrm{WN} | x_\mathrm{n},x_\mathrm{p} \rangle=0$ (Inset Fig.~\ref{fig_Kxp}a), and we directly recover the expected contractile elastic coupling (Eq.~\eqref{eqn_protrusion}) in the PVM (Fig.~\ref{fig_Kxp}a). In clear contrast to this prediction, the PVM inferred from experiments shows an intricate dependence of the protrusion velocities as a function of $x_\mathrm{n}$ and $x_\mathrm{p}$ (Fig.~\ref{fig_Kxp}c). These results indicate that to account for the experimentally observed dynamics, we need to account for time correlations in the polarity (Appendix~\ref{sec_protrusion}).

The overall structure of the experimental PVM is in line with a contractile coupling between nucleus and protrusion: it exhibits negative velocities for $x_\mathrm{p}>x_\mathrm{n}$ and positive velocities for $x_\mathrm{p}<x_\mathrm{n}$. These features correspond to the protrusion being pulled back towards the nucleus. However, when the protrusion extends into the constriction, the protrusion velocity switches sign, corresponding to an unexpected driving force pushing the protrusion away from the nucleus. This `polarity driving' cannot be accounted for even by a non-linear contractile coupling to the nucleus in our model with a white-noise polarity. Instead, we expect that the polarity may exhibit time-correlations to account for the spatiotemporal dynamics of the migration machinery.

To investigate how time-correlated polarity dynamics affect the migration behavior, we consider the simplest choice of a persistent, exponentially correlated polarity
\begin{equation}
\label{eqn_POU}
\dot{P}_\mathrm{per} = - \alpha_0 P_\mathrm{per} + \sigma \xi(t)
\end{equation}
with $\alpha_0>0$. In this case, the polarity experiences negative feedback, $\dot{P}_\mathrm{per} \propto -P_\mathrm{per}$, and exhibits time-correlations $\langle P_\mathrm{per}(0) P_\mathrm{per}(t) \rangle$ decaying exponentially on a persistence time-scale $\alpha_0^{-1}$. These polarity dynamics have significant correlations with the state of the system: $\langle P_\mathrm{per} | x_\mathrm{n},x_\mathrm{p} \rangle \neq 0$ (Inset Fig.~\ref{fig_Kxp}b), and thus, unlike in the white-noise case, the polarity contributes to the PVM (Eq.~\eqref{eqn_PVM}). Specifically, the persistent polarity exhibits a polarity driving similar to the experimental PVM (Fig.~\ref{fig_Kxp}b,c). Taken together, these results indicate that cell protrusions are driven by time-correlated polarity dynamics.

\begin{figure}[h!]
	\includegraphics[width=0.45\textwidth]{fig5}
	\centering
		\caption{
		\textbf{Protrusion velocity maps and the geometry adaptation model.}
		\textbf{a,b.} Protrusion velocity maps (PVM) $\langle \dot{x}_\mathrm{p} | x_\mathrm{n},x_\mathrm{p} \rangle$ predicted by the white noise model, and the persistent polarity model. In both models, we use a potential to enforce the overall system boundaries, $V(x_\mathrm{p}) = (x/x_\mathrm{boundary})^8$ (Appendix~\ref{sec_protrusion}).
		\emph{Inset:} polarity contribution to the PVM, given by $\langle P | x_\mathrm{n},x_\mathrm{p} \rangle$.
		\textbf{c.}  PVM inferred from experiments with bridge width $W=7 \ \si{\micro\meter}$.
		\textbf{d.} Schematic illustration of the polarity models. The persistent polarity corresponds to a flat feedback profile, $\alpha(x_\mathrm{p})=\alpha_0$, where the polarity is effectively confined to a constant harmonic potential $U(P)$ (left). In the geometry adaptation model (Eq.~\eqref{eqn_Kxp}), $\alpha_\mathrm{min}$ controls the sign of the feedback. For $\alpha_\mathrm{min}>0$, the feedback is negative and the polarity is confined to harmonic potentials with spatially varying stiffness (left). For $\alpha_\mathrm{min}<0$, the feedback locally becomes positive, leading to two stable fixed points (right).
		\textbf{e.} From left to right, we vary the value of $\alpha_\mathrm{min}= \{6,4,0,-4,-6\} \ \mathrm{h}^{-1}$. The joint probability distribution of protrusion position and polarity $p(x_\mathrm{p},P)$ is shown. Solid red lines indicate the position of the stable fixed points in the polarity dynamics, dashed red lines indicate unstable fixed points.
		\textbf{f.} Joint probability distributions $p(x_\mathrm{n},x_\mathrm{p})$ predicted by the geometry adaptation model with varying $\alpha_\mathrm{min}$.
		\textbf{g.} Probability distribution of the dwell times $\tau$ predicted by the model (red) and observed experimentally ($W=7 \ \si{\micro\meter}$, blue).
		 }
	\label{fig_Kxp}
\end{figure}

\subsection*{Confinement triggers polarity self-reinforcement}

While the persistent polarity (Eq.~\eqref{eqn_POU}) describes the qualitative features of the protrusion velocities (Fig.~\ref{fig_Kxp}b,c), it predicts stochastic dynamics that do not capture the key features of the experiment. Specifically, it fails to capture the stereotypical protrusion-nucleus cycling indicated by the ring structure in the experimental probability distribution $p(x_\mathrm{n},x_\mathrm{p})$ (Supplementary Fig. S4). The persistent polarity relies on several simplifying assumptions. Firstly, we assumed the polarity to be insensitive to the local confinement, as the polarity dynamics do not explicitly depend on the position of the protrusion. Secondly, we assumed negative feedback, $\dot{P}_\mathrm{per} \propto -P_\mathrm{per}$. This means that the polarity is effectively confined to a harmonic potential $U(P_\mathrm{per})= \alpha_0 P_\mathrm{per}^2 /2$, and thus always driven back towards zero (Fig.~\ref{fig_Kxp}d).

To relax these assumptions, we propose a geometry adaptation (GA) model, where the strength and sign of the polarity feedback depend on the local geometry of the confinement. Thus, the feedback may vary with the position of the protrusion, $\alpha=\alpha(x_\mathrm{p})$. Physically, we expect that the polarity may become more persistent when the protrusion is in the constriction. Such an increase in persistence could be due to increased alignment of actin fibers~\cite{SoaresESilva2011,Bonelli2016,Jiang2005,Prager-Khoutorsky2011}, or more stable polarity cue gradients~\cite{CallanJones2016,Maree2012a,Vasilevich2021} when the protrusion is confined to a narrow constriction. To ensure that the polarity remains bounded, we include the next-order term allowed by symmetry $-\beta P^3$, with $\beta>0$, and allowing $\alpha<0$ locally:
\begin{equation}
\label{eqn_Kxp}
\dot{P}_\mathrm{GA} = - \alpha(x_\mathrm{p}) P_\mathrm{GA} -\beta P_\mathrm{GA}^3  + \sigma \xi(t)
\end{equation}
To account for larger persistence in the constriction, we choose a feedback function $\alpha(x_\mathrm{p})$ with a minimal value $\alpha_\mathrm{min}$ in the center of the constriction (Fig.~\ref{fig_Kxp}d). If $\alpha_\mathrm{min}>0$, the polarity dynamics exhibits a stable fixed point at $P_\mathrm{GA}=0$ everywhere (Fig.~\ref{fig_Kxp}d). In contrast, if $\alpha_\mathrm{min}<0$, the polarity is still driven back to $P_\mathrm{GA}=0$ on the islands, but in the constriction, two stable fixed points $P_\mathrm{GA}^*= \pm \sqrt{|\alpha|/\beta}$ appear. Consequently, when the protrusion is in the constriction and the polarity is small ($|P_\mathrm{GA}|<|P_\mathrm{GA}^*|$), a positive feedback mechanism $\dot{P}_\mathrm{GA} \propto P_\mathrm{GA}$ is activated, leading to a self-reinforcement of the polarity in the current direction of polarization, breaking the symmetry of the dynamics.

We now explore the predictions of the geometry adaptation model by varying $\alpha_\mathrm{min}$ (Fig.~\ref{fig_Kxp}e). As expected, for negative polarity feedback ($\alpha_\mathrm{min}>0$), we find a polarity distribution $p(x_\mathrm{p},P)$ with polarities centered around $P=0$ at all positions $x_\mathrm{p}$. In contrast, for positive feedback, polarities preferentially take finite values in the constriction, yielding a ring-structure in the polarity distribution $p(x_\mathrm{p},P)$ (Fig.~\ref{fig_Kxp}e). The model then predicts protrusion-nucleus cycling in $p(x_\mathrm{n},x_\mathrm{p})$ and a marked peak in the dwell time distribution, defined as the time between subsequent transitions of the cell nucleus across the bridge centre, both in line with the experiment (Fig.~\ref{fig_Kxp}f,g).

Up to this order of description, there are two possible alternative models allowed by symmetry which couple the polarity to the state of the system. First, instead of coupling the polarity feedback to the geometry, the overall amplitude of the polarity could depend on position. Second, instead of a position-dependent feedback, one could consider feedback that depends on the cell's extension $|x_\mathrm{n}-x_\mathrm{p}|$. However, these alternatives fail to capture our experimental observations (Appendix~\ref{sec_protrusion}). Taken together, these results suggest that to capture the stereotypical protrusion-nucleus cycling, we require a geometry-sensitive polarity feedback.

\begin{figure}[h!]
	\includegraphics[width=0.48\textwidth]{fig6}
	\centering
		\caption{
		\textbf{Geometry adaptation model predicts dynamics with varying constriction width.}
		\textbf{a.}	Stochastic trajectories $x_\mathrm{n}(t)$ (blue), $x_\mathrm{p}(t)$ (green), and $P_\mathrm{GA}(t)$ (pink) predicted by our mechanistic model, which combines the adhesion landscape and geometry adaptation (Eq.~\eqref{eqn_adhesion},~\eqref{eqn_protrusion},~\eqref{eqn_Kxp}).
		We simultaneously increase $\gamma_\mathrm{min}$ and $\alpha_\mathrm{min}$ to account for increasing bridge width $W$ (Appendix~\ref{sec_protrusion}).
		\textbf{b.} Joint probability distributions $p(x_\mathrm{n},x_\mathrm{p})$.
		\textbf{c.} Protrusion velocity maps (PVM) $\langle \dot{x}_\mathrm{p} | x_\mathrm{n},x_\mathrm{p} \rangle$. The top row corresponds to the model prediction, the bottom row to experimental observations.
		\textbf{d.} Predicted (red) and experimental (blue) dwell time distributions $p(\tau)$.
		\textbf{e.} Predicted (red) and experimental (blue) effective friction at the bridge center $F(x_\mathrm{n} \to 0,v_\mathrm{n})$.
		In all panels, experimental observations correspond to $W=4,7,12,22,35 \ \si{\micro\meter}$ (from left to right). For additional quantifications, see Supplementary Fig. S8.
		}
	\label{fig_w}
\end{figure}

\subsection*{Geometry adaptation model correctly predicts response to varying constriction dimensions}

Having constrained the model based on a single confining geometry, we challenge the predictive power of our approach by investigating the full stochastic trajectory dynamics of cells in micropatterns with varying constriction geometry. Specifically, we have fully constrained the model parameters for the adhesion landscape model for the nucleus (Eq.~\eqref{eqn_adhesion}; Fig.~\ref{fig2}) and the geometry adaptation model for protrusion and polarity (Eq.~\eqref{eqn_protrusion},~\eqref{eqn_Kxp}; Fig.~\ref{fig_Kxp}). Increasing constriction width has a clear implication for this model: in addition to a flattening adhesiveness profile $\gamma(x_\mathrm{n})$ (Fig.~\ref{fig3}a), we also expect the positive polarity feedback to diminish. The model predicts that the protrusion-nucleus cycling should disappear with increasing constriction width: the ring-structure of the position distribution $p(x_\mathrm{n},x_\mathrm{p})$ gradually closes (Fig.~\ref{fig_w}b), and the typical transition time-scale, indicated by the peak in the dwell time distribution, disappears (Fig.~\ref{fig_w}d). Furthermore, this mechanistic model explains the nonlinear dynamics observed in the underdamped dynamics of the cell nucleus alone~\cite{Brueckner2019}, including the effective `negative friction' at the center of the constriction (Fig.~\ref{fig_w}e, Appendix~\ref{sec_xv}). All these predicted features of the dynamics are quantitatively confirmed experimentally (Fig.~\ref{fig_w}b-d). Importantly, these predictions involve no further fitting: the key parameters $\{ k_\mathrm{n}, \gamma_\mathrm{min}, k_\mathrm{p}, \alpha_0, \alpha_\mathrm{min} \}$ for a thin constriction ($W=7 \ \si{\micro\meter}$ are completely determined in Fig.~\ref{fig2}, \ref{fig_Kxp}, and the variation of $\gamma_\mathrm{min},\alpha_\mathrm{min}$ with increasing width is determined by the micropattern geometry (Appendix~\ref{sec_protrusion}).

These observations can be understood by examining the protrusion dynamics in the model: we find that the predicted PVM exhibits a polarity driving of decreasing magnitude with increasing bridge width (Fig.~\ref{fig_w}c). Notably, the driving disappears in the widest system with no constriction, where the protrusion velocities are thus determined by the protrusion-nucleus coupling. This coupling pulls protrusion and nucleus together, inhibiting the stereotypical cycle. The model therefore suggests that these stereotypical cycles rely on the adaptation of the cell polarity dynamics to its local environment. We further challenge our model by exploring geometries with varying constriction length $L$, and the inclusion of multiple adhesive islands to create multi-state micropatterns (SI Movie S8). We find that the model also captures the changes in dynamics observed in these systems (Appendix~\ref{sec_geom}). Taken together, these results indicate that our mechanistic model has strong predictive power beyond the specific confinement geometry which we used to constrain it.

\begin{figure*}[]
	\includegraphics[width=\textwidth]{fig7}
	\centering
		\caption{
		\textbf{Geometry adaptation model captures response to pharmacological perturbation of polarity, contractility, and actin polymerization.}
		\textbf{a.} Feedback profiles $\alpha(x_\mathrm{p})$ for varying strengths of the positive feedback, showing $\alpha_\mathrm{min}=\{-6,-4,-2,0\}$.
		\textbf{b.} Model trajectories of $x_\mathrm{n}(t),x_\mathrm{p}(t)$ for the standard value $\alpha_\mathrm{min}=-6$, corresponding to blue curves in panels a,d,e; and $\alpha_\mathrm{min}=-2$, which provides a model for reduced polarity feedback. The adhesion profile is held constant, corresponding to no changes in the confinement geometry.
		\textbf{c.} Corresponding probability distributions $p(x_\mathrm{n},x_\mathrm{p})$.
		\textbf{d.} Dwell time distributions $p(\tau)$.
		\textbf{e.} Effective friction at the bridge center $F(x_\mathrm{n} \to 0,v_\mathrm{n})$.
		\textbf{f.} Experimental trajectories of $x_\mathrm{n}(t),x_\mathrm{p}(t)$ on micropatterns with constriction width $W=7\si{\micro\meter}$, for the control condition, treatment with the ROCK-inhibitor Y-27632, blebbistatin, and latrunculin.
		\textbf{g.} Corresponding probability distributions $p(x_\mathrm{n},x_\mathrm{p})$.
		\textbf{h.} Dwell time distributions $p(\tau)$.
		\textbf{i.} Effective friction at the bridge center $F(x_\mathrm{n} \to 0,v_\mathrm{n})$.
		}
	\label{fig_drugs}
\end{figure*}

\subsection*{Geometry adaptation depends on cell polarity, contractility and actin polymerization}

Key processes that affect the state of cell polarization are the structure and contractility of the actin network~\cite{Jiang2005,Prager-Khoutorsky2011,Trichet2012,Gupta2015,Ladoux2016}, as well as diffusable polarity cues~\cite{CallanJones2016,Hodge2016,Warner2019}, including Rho GTPase. To test whether geometry adaptation depends on these cellular components, we pharmacologically interfered with cell polarity, contractility, and protrusion formation. Specifically, we use inhibitors of Rho-associated protein kinase (ROCK) (Y-27632), myosin-II (blebbistatin) and actin polymerization (latrunculin) (Fig.~\ref{fig_drugs}f, SI Movies S9-11). Interestingly, in the case of ROCK and myosin inhibition, we observe increased probability in the center of the ring-like probability distribution $p(x_\mathrm{n},x_\mathrm{p})$, a disappearance of the peak in the dwell time distribution, and a reduction of the negative friction in the nonlinear nucleus dynamics (Fig.~\ref{fig_drugs}g-i). The effect of actin polymerization inhibition are less pronounced, but qualitatively similar. This set of observed changes to the dynamics are in congruence with the predictions of our model following a reduction of the feedback strength $\alpha_\mathrm{min}$ (Fig.~\ref{fig_drugs}a-e). Importantly, changing other aspects of the model, such as the adhesion landscape, cannot capture this set of trends (Supplementary Fig. S10). Thus, these results suggest that perturbation of cell polarity, myosin contractility and actin polymerization reduces the strength of the geometry adaptation, indicating that the geometry adaptation mechanism depends on these cellular components.

\subsection*{Discussion}

In this work, we develop a mechanistic theoretical framework to describe the joint stochastic dynamics of cell nucleus, protrusion, and polarity, and their coupling to the extracellular microenvironment. Experimentally, we find that cells migrating in confinements with a thin constriction exhibit a stereotypical protrusion-nucleus cycling, with characteristic protrusion growth followed by a rapid transition of the nucleus across the bridge. Using a large experimental data set of joint protrusion and nucleus trajectories, we systematically constrain a mechanistic model for confined cell migration.

In our model, we identify three distinct stages of the protrusion-nucleus cycling (Fig.~\ref{fig_sum}). First, we observe an initial exploration phase, where both nucleus and protrusion are located on the same island (Stage I). At this stage, the polarity is subject to negative feedback, causing the protrusion to frequently change direction and explore its surroundings. Stochastic polarity excitations can trigger the protrusion to enter the constriction. Within the constriction, the protrusion becomes highly confined, causing the polarity dynamics to switch from a negative to a positive feedback loop. This positive feedback reinforces the polarity, driving the protrusion growth into the constriction (Stage II). At the same time, tension builds up due to the coupling to the nucleus, which is held back on the island due to the enhanced adhesion with the substrate. Once the protrusion reaches the other end of the system, the nucleus is pulled across the constriction, relaxing the tension in the elastic coupling, reminiscent of a slingshot (Stage III). The three stages of the transition process arise as a consequence of the interplay of the three key physical mechanisms in the system: the adhesion landscape, the nucleus-protrusion coupling, and the polarity self-reinforcement.

To develop our theoretical approach, we separately constrain the dynamics of the nucleus and the protrusion and systematically consider model terms of increasing complexity (Table~\ref{tab:model_table}). We first studied the stochastic dynamics of the cell nucleus. Interestingly, the nucleus dynamics are inconsistent with movement in a simple double-well potential, as might be expected for example from cellular deformation arguments~\cite{Albert2014,Bi2014,Bi2015,Segerer2015,Goychuk2018}. Indeed, active particles confined to double-well potentials can exhibit excitable dynamics similar to those observed in the experimental trajectories of the nucleus alone~\cite{Caprini2019}, making the double well a promising model candidate. However, based on the observed joint dynamics of nucleus and protrusion, we find that this energy barrier model is unable to capture our experiments.

Instead, our model suggests that the movement of the nucleus is determined by the locally available adhesive area, manifesting as an adhesion landscape with a spatially variable friction coefficient. Thus, as the protrusion explores the environment, the back of the cell `sticks' due to the high adhesiveness on the island. This is in line with experimental observations showing that in mesenchymal migration, the movement of the cell rear, where the nucleus is typically located, is limited by the unbinding of mature adhesions~\cite{Gupton2006,Giannone2009}. In the constriction, the cell polarity actively drives the protrusion away from the nucleus, causing mechanical stress to build up in the protrusion-nucleus coupling. This `self-loading' of the coupling eventually causes a contraction stage, where the cell quickly contracts and the nucleus rapidly moves across the constriction. In the model, the tension in the elastic coupling rapidly relaxes during the slip phase, similar to a slingshot. Such `slingshot' dynamics have also been observed in confined 3D migration in fibrous matrices~\cite{Wang2019a}.

The adhesion-limited nucleus motion is reminiscent of stick-slip processes that have been observed in cell migration on 1D lines~\cite{Monzo2016,Hennig2020,Ron2020a} due to the mechanosensitive binding and unbinding dynamics of adhesions~\cite{Ron2020a,Sens2020}. In contrast, our work suggests that a key determinant for  stick-slip dynamics in confined systems is the interplay of the geometry-sensitive polarity dynamics with the elastic protrusion-nucleus coupling, leading to the self-loading of the coupling. Specifically, our model indicates that the polarity dynamics adapts to the presence of the constricting geometry by activating a self-reinforcing positive feedback loop. This positive feedback leads to a broken-symmetry state, in which there is a non-zero preferred polarity. Previous work on unconfined 2D migration has suggested a similar symmetry breaking mechanism in the cell velocity dynamics, leading to a long-lived state of spontaneous polarization~\cite{Maiuri2015}. However, in that case, this state emerged for particular cell parameters. In contrast, our work suggests that such states may also arise as a consequence of adaptation to a confinement.

We were able to rule out an alternative model to the position-dependent feedback where the polarity dynamics do not depend explicitly on external geometry, but on the extension of the cell (Appendix~\ref{sec_protrusion}). An interesting aspect of comparing these two models are their conceptually distinct implications. The position-dependent feedback implies a direct coupling to the external environment, where the cell may sense and adapt to the external geometry. In contrast, an extension-dependent feedback is translationally invariant, and only depends on the internal state of the cell. Such a mechanism has recently been suggested to be decisive for protrusion growth in keratocytes~\cite{Raynaud2016a}. In contrast, our findings suggest that the positive feedback loop in the polarity is a response to the geometry of the local microenvironment rather than to the overall extension of the cell.

We demonstrate that the geometry adaptation of protrusion and polarity dynamics suggested by our model depends on several underlying biological mechanisms, including the polarity-mediating Rho-associated protein kinase (ROCK), myosin contractility, and actin polymerization. This observation suggests that geometry adaptation may be controlled by the underlying polarization mechanisms of the cell. There are a several ways these mechanisms could contribute to the geometry adaptation of cell polarity dynamics. First, based on the physics of active gels, which describe, for example, the actomyosin cortex in the protrusion, we expect a greater degree of alignment of actin fibers in a narrow constriction~\cite{SoaresESilva2011,Bonelli2016}. Increased alignment of actin is associated with higher myosin-contractility~\cite{Prost2015,Julicher2018} and the emergence of spontaneous cell polarization~\cite{Jiang2005,Prager-Khoutorsky2011,Trichet2012,Gupta2015,Ladoux2016}. A further key determinant of cell polarization are diffusable polarity cues, such as Rho GTPase~\cite{CallanJones2016,Hodge2016,Warner2019}, whose spatiotemporal organization may couple to external geometries, for example through focal adhesions~\cite{Demali2003}, or the cell shape itself~\cite{Maree2012a,Vasilevich2021}. By combining the data-driven mechanistic modelling developed in this work with cytoskeletal perturbations and imaging, the biological and molecular underpinnings of geometry adaptation could be further elucidated in future work.

Based on experiments in which we varied the dimensions of the constriction as well as the number of adhesive islands to create multi-state micropatterns (Appendix~\ref{sec_geom}), we found that our model has predictive power beyond the specific confinement geometry used to constrain it. Thus, this model could potentially be used to make predictions for new experiments, such as cell migration on patterned lines~\cite{Schreiber2016,Schreiber2021,Caballero2014,Caballero2015}, in 3D-confinements~\cite{Patteson2019,Davidson2020}, or at junctions in a maze~\cite{Renkawitz2019}. Previous work has investigated the effect of asymmetric periodic patternings, which led to a rectification of cell migration in one direction~\cite{Mahmud2009,Caballero2014,Caballero2015}. This rectification has been interpreted to be a consequence of the asymmetry in locally available adhesive area~\cite{LoVecchio2020}, consistent with our adhesion landscape model. Our work suggests that the adaptation of cell polarity in response to confinements may also play an important role in such processes. Furthermore, protrusion and polarity dynamics are critical in migration in 3D extra-cellular matrices~\cite{Friedl2003,Fraley2010,Caswell2018}, as well as in pair-wise interactions of cells~\cite{Abercrombie1953,Carmona-Fontaine2008,Brueckner2021,Zisis2022,LaChance2022}, which in turn control the collective dynamics of cells~\cite{Alert2020}. The geometry adaptation dynamics we have identified here could therefore play an important role in these more complex processes, and provide a new framework for physical models of cell migration in confining systems.

\subsection*{Author Contributions}
D.B.B., E.H. and C.P.B. conceived the project; A.F., D.B.B., C.P.B. and J.R. designed experiments; A.F. and G.L. performed experiments; A.F., G.L. and N.A. performed tracking; M.S., J.F. and D.B.B. developed the image segmentation; D.B.B. and M.S. analysed data; D.B.B., M.S. and C.P.B. developed the theory; D.B.B. and C.P.B. wrote the manuscript with input from all authors. \\\

\subsection*{Acknowledgements}
We thank Grzegorz Gradziuk, Steven Riedijk, Janni Harju, and Schnucki for helpful discussions, and Andriy Goychuk for advice on the image segmentation. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 201269156 - SFB 1032 (Project B12). D.B.B. is a NOMIS fellow supported by the NOMIS foundation and was supported in part by a DFG fellowship within the Graduate School of Quantitative Biosciences Munich (QBM) and by the Joachim Herz Stiftung.

redpajama_set_name: RedPajamaArXiv


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