Thomas Piketty

Rémi Jardat
email: r.jardat@istec.fr

Prof Rémi Jardat
email: remi.jardat@centraliens.net

"For Piketty"a new Enlightenment that goes far beyond rewriting Das Kapital Le capital au XXIe Siècle (Capital in the Twenty-First Century) by

Keywords: Piketty, Capital, Marxism, Enlightenment, Rosanvallon, Wallerstein

conducts research in various fields of

Introduction

Thomas Piketty's latest opus magnum, Capital in the Twenty-First Century, is certain to have a considerable impact on economic and social science scholarship for many years to come. At the time of this writing, it has garnered overwhelmingly favorable reviews from scholars of the ilk of Jeremy Rifkin and Nobel Prize laureate Paul Krugman, as well as earning accolades on the pages of highly regarded economic newspapers such as The Economist.

Reading this quite long but never boring 700-some-page tome (if you include the extensive footnotes) delivers a forceful jolt. In a word, the author contends that today's world and the likely near future look radically different from what we have envisaged heretofore. And that radical shift in perception is supported by rigorously documented data. What is more, the author develops a few secondary arguments that are of great importance for scholars, especially those in the field of management science and corporate governance.

In the long term, Piketty's Capital will surely have the same impact as recent, acclaimed Marxist and -Marxistoïd‖ literature, and it may even demand comparison with Karl Marx's famous work. Yet, although Piketty addresses certain issues developed by Marx (which may explain some heated criticism from left-leaning reviewers) his ideas are anything but Marxist. Far from being a skeptical philosopher out to vent his frustration, Piketty is perhaps best seen as undertaking an act of new Enlightenment. Notably, he offers the grounds for a serious support to the French leftist scholarship laid out in the works of Pierre Rosanvallon and propagated through Rosanvallon's now decade-long editorship at the intellectual publishing workshop, La République des Idées (Editions du Seuil, Paris).

It is most fortunate that this book was based, in large part, on studies using statistical data gathered on France and the US. For not only do Piketty's arguments find resonance in the general public and in academic circles on both sides of the Atlantic, they also speak to cultures where the notions of equality and Enlightenment are deeply anchored, due to both nations' revolutionary past and their claim to embody -universal‖ values. However, international scholars may fail to fully grasp the importance of Piketty's theses to somewhat Franco-centric debates. Hence the perspective through which we have chosen to discuss Piketty's work in this paper.

First, we shall present an overview of Piketty's ideas and then briefly elaborate on the import of his claims and economic observations. Secondly, we shall focus on what is no doubt the book's most thought-provoking argument: after describing the grand dynamic toward the accumulation of capital, the author proposes consequences that challenge not only modernday perceptions of wealth creation and distribution, but also our conception of what life is and how it should be lived in democratic, merit-based societies. Thirdly, we intend to focus on Piketty's strong epistemological stances, while placing them in the context of Marxist thought. On the basis of these three points, we shall argue, lastly, that Piketty is more a Philosopher of a New Enlightenment than the latter-day Marxist that some make him out to be. At bottom, he considers the fundamental question of -Who are we?‖ that occupies the mind of most every man endowed with reason; but Piketty also invites us to revisit our inquiry through Foucault's -sagittal gaze‖ (1994), as a fundamental problem of our time.

[Texte] 15/04/2014 1.

Piketty's main thesis: the inexorable dynamic of capital accumulation--and concentration

Piketty should be seen as not merely a theorist of inequality but also as the purveyor of a new theory of capital and wealth, who has bolstered his new, dual status with solid empirical data. His theory describes the mechanisms by which the growth rate and the increasing concentration of capital are both spontaneous and not moderated by natural market forces. By painstakingly charting income and wealth statistics over long spans of time, he has provided empirical evidence that gives the lie to the illusion that rent and inherited wealth would eventually fade away in favor of income earned from labor and merit-based talent. This is, as we shall see, an illusion built on two main sources.

The illusion of meritocratic hope

The first source is the very specific historical and demographic parameters that characterize the 20th Century--a century that witnessed the destruction of the stock of capital during the two world wars, dramatic population growth, and the subsequent rejuvenation of the economy. Piketty provides incontestable evidence that, despite affording a certain comfort to academics, statistical findings on wealth and capital accumulation and distributions (especially as derived from Solow's model or Kuznets' curve) are skewed because data is read across a limited, exceptional period of time. But now that those particular historical chapters are closed, it is very likely that the 21st century will see the renewed accumulation of capital: national outputs should tend to equalize worldwide, as the sum of the growth rates of population and productivity both diminish and stabilize, with more and more countries catching up with the Technological Frontier, so that growth cannot surpass 1%-2% per year. As Piketty explains, weak, stable growth at such rates is different from secular stagnation. It is a situation that can potentially produce deep, beneficial effects over the long run and foster social and technological change. However, if no deliberate institutional or, more likely, public policies are devised to counter the concentration of capital and limit inequality, the trend will continue unabated and attain such levels by century's end that one might legitimately wonder if the world is owned by a handful of extremely wealthy private individuals and institutions. For there is every chance that, by the end of the century, 99.99% of us will be working for the Qatari emir or a few Chinese oligarchs.

The second source of illusion resides with methods of data analyses. That is, the statistical tools and surveys normally used to measure economic inequalities are, in Piketty's view, totally inadequate for charting the inexorable, dramatic shift toward a heavy concentration of income and, mostly importantly, wealth, in the upper echelons of an economic elite. Wealth is becoming increasingly concentrated among those at the very top end of the income scale but the extent of that fact is overlooked or under-estimated in traditional indexes, essentially because they tend to focus on income and wealth brackets calculated only in deciles, whereas a breakdown in the first centile and, eventually, each millime, would yield a clearer picture of wealth concentration.

The unstoppable mechanism driving capital and wealth

By relying on more reliable and complete statistical data on wealth distribution, taking the form, essentially, of tax and estate tax/inheritance records going far back into the past, Piketty illustrates that his very simple model of the distribution of income between capital and labor, and the distribution of wealth and income, can stand not only as an elegant theoretical [Texte] 15/04/2014 construct but also as a highly reliable indicator and predictor of socio-economic conditions. Given that all these aspects of his methodological framework are well documented in the book and the associated website, we shall focus, here, only on the key aspects of Piketty's discussions on the dynamics of income and capital, as elaborated in Parts I and II. We can summarize those dynamics as follows:

 Two key variables -Income‖ and -Capital‖ are examined in an approach that marks a slight shift in the way such parameters are generally used in modern economic debates:

-National income is defined as the sum of all income available to the residents of a given country in a given year, regardless of the legal classification of that income‖ Piketty explains (p. 43).1 It should be noted that he calculates national income as Gross Domestic Product (GDP) minus capital depreciation, which measures the loss in capital value owing to the use of capital goods in production (yielding domestic output) plus net income from abroad. Whereas most analysts focus on GDP, Piketty shifts attention to the variables making it possible to account for the sharp disparities between peoples' economic condition (i.e. those who live off labor income compared to those who live off rental income, dividends and inherited wealth) as well as between nations (i.e. countries that differ according to their capital labor ratios or their status as donor or recipient of assistance to meet funding requirements). In this way, Piketty not only captures the true picture of the economic and social forces driving the relationships between people and nations, he succeeds in identifying a method for measuring disparities that people sense viscerally but have often found difficult to clearly articulate.

He provides an equally explicit and meaningful definition of the economic concept of capital: -In this book, capital is defined as the sum total of nonhuman assets that can be owned and exchanged on a market. So, whereas the term -capital‖ includes all forms of real property (including residential real estate) as well as financial and professional capital (plants, infrastructures, machinery, patents, and so on) used by firms and government agencies‖ (p. 46), it is important to note that Piketty's definition excludes so-called -human capital‖ (considering that, since the abolishment of slavery, no one can be actually be owned or traded). He also sees no reason to make a distinction between capital and wealth. Here, he puts his finger on an all too often neglected part of wealth: real estate, land, or property, if you will. The point is, in contrast to neoclassical economists who have generally relegated land to irrelevance in their equations, Piketty places it at the center of his calculations, while reminding us that about half of the capital in our societies now consists precisely of landbased capital--agricultural land as well as buildings and housing values. As Piketty emphasizes, -capital stock in the developed countries currently consists of two roughly equal shares: residential capital and professional capital used by firms and government.‖ (p. 51). As we shall see below, that second shift in variable is of major importance.

 Two key ratios: , the share of capital in national income (which gives us a measure of the split between income from work and income from wealth) and , the ratio of capital to income (usually expressed in years of national income).

As regards , the key point to bear in mind is its rise observed in the past three decades, in Europe as well as in the US. This means that, in our societies, owners of capital [Texte] 15/04/2014 capture an ever-expanding share of the value created by workers. The crucial issue for the 21st century is that  is very likely to rise at levels rivaling or even surpassing those witnessed in the Belle Epoque (or the Gilded Age) just before World War I. This deals a serious blow to the notion commonly accepted among the labor classes, in particular, and meritocratic societies, in general: namely, that gaining a sense of accomplishment in life and work depends on individual effort and achievement. Piketty insists on this point and uses it to pose a compelling question: -Is it useful and just for the owners of capital to receive this marginal product [of capital] as payment for their ownership of property (whether their own past savings or that of their ancestors) even if they contribute to no new work?‖ (p. 215). All other things being equal,  is all the more likely to rise as the stock of capital expands compared with the stream of national income, that is, when measured in terms of the ratio  of capital to national income. The irrefragable fact is that , which was very low after the eventful decades of the 20th Century, is now quite high: capital now typically attains a level of 5 to 6 years' worth of national income in the developed countries. On the one hand, accumulating a huge amount of wealth in our countries may be considered to constitute an unequivocal success. Moreover, it renders the problem of debt potentially quite easy to solve, insofar as debt corresponds to barely one year of income in the West. However, this situation has an alarming underside: it means that only a small part of wealth is renewed every year, which tends to favor domination by those who are able to accumulate wealth (the rich or the elderly) over younger, working people. As might be expected, this affects how societal dynamics play out. The fact that accumulated capital is mostly held in private hands and is extremely concentrated makes the situation even more problematic. Moreover, it is illusory to think that technology would mitigate that dynamic. In Piketty's words: -Modern technology still uses a great deal of capital, and even more important, because capital has many uses, one can accumulate enormous amounts of it without reducing its return to zero. Under these conditions, there is no reason why capital's share must decrease over the very long run, even if technology changes in a way that is relatively favorable to labor.‖ And so it is facetious to imagine that -fat cat stockholders have supposedly been replaced by talented managers thanks solely to changes in technology‖ (p. 224).

 Two momentums of change: net return on capital, r, and growth in output, g.

Capital undergoes its own dynamic. Significantly, r, the return on capital, stabilizes on average at a rate of 5%-6% in a normal economy. By contrast, the dynamic of output and revenue is lower. In every country where the frontier of technological innovation has been reached, neither inflation nor growth g should exceed 1.5%-2%. That means that, once a given amount of wealth has been accumulated, -People with inherited wealth need save only a portion of their income from capital to see that capital grow more quickly than the economy as a whole. Under such conditions, it is almost inevitable that inherited wealth will dominate wealth amassed from a lifetime's labor by a wide margin, and the concentration of capital will attain extremely high levelslevels potentially incompatible with the meritocratic values and principles of social justice fundamental to modern democratic societies‖ (p. 26).

That positive feedback process underlies the relentless forces that drive the formation of capital in a period of economic stability, as was the case during the 19th century -Belle Époque‖ France and Edwardian Britain until World War I. It is a process that engenders sharp disparities whereby the wealthy become more and more disconnected from the rest of society. At the other end of the class spectrum, those born without any capital-that is, without landbased, industrial or finance capital-have no chance of becoming members of the economic elite, because growth is too low to provide opportunities to accumulate such levels of wealth, even though individuals relying solely on wages and salaries generate resources and make [Texte] 15/04/2014 payments that contribute to wealth accumulation by the wealthiest, whether in the form of the rent paid directly to owners for housing, or through tax payments, from which the wealthiest benefit, indirectly, when such sums are applied to repay state debt to its main creditors, namely, the wealthiest!

The contrasting paths taken by the capital-rich and the capital-poor are all the more divergent given that, as statistical evidence suggests, the net return on capital, r , increases in correlation to the level of wealth enjoyed by the individual, mainly because the wealthy can take more risks with their savings and can earn higher returns. To give full expression to the perversity of this dynamic, Piketty resorts to language accentuating capital's brutal, unrelenting qualities: -Capital is never quiet: it is always risk-oriented and entrepreneurial, at least at its inception, yet it always tends to transform into rents as it accumulates in large enough amounts -that is its vocation, its logical destination.‖ (pp. 115-116).

 A simple equation  = r x , combined with the asymptotic empirically proven law  = s/g, These equations give insight into the macroeconomic effects of the positive feedback between capital and rent described above. The first equation  = r x expresses the fact that the share of capital in national income is the product of the return on capital, r, multiplied by the capital/income ratio. It is already known that r is going to remain at a high and stable level of 4.5%-5%. The second equation predicts that  is very likely to remain high, too, in a future period of high savings (s) and low growth (g).

In light of the foregoing, the share of capital in national income should become higher and higher in the long run: -In a quasi-stagnant society, wealth accumulated in the past will inevitably acquire disproportionate importance‖ (p. 166). That means that capital is going to capture a bigger and bigger share of the national income, compared with labor. And so a question arises: -Who are our children going to work for?‖ Most likely not for themselves, but, to a growing extent, for wealth holders who have amassed their fortunes by virtue of heredity rather than based on merit. Piketty views that economic outlook as involving a sort of historical involution: -By 2100, the entire planet could look like Europe at the turn of the twentieth century, at least in terms of capital intensity‖ (p. 196).

 Two empirical statements over the long run, combined with very likely outlook: r-g>2.5% and 1.3< (where  is the elasticity of the substitution of capital for labor in the output function).

The variables employed in the above equations are not to be seen as independent. More specifically, Marx and other scholars of the 20th century had stated that return on capital should be naturally moderated as the stock of capital grows. This reflected a very basic law of supply and demand applied to the capital market and capital cost. However, as Piketty shows, even though r may slightly decrease over the long run, the effect of capital accumulation should be so great that the split between income from work and income from wealth would tend to favor capital at the expense of labor. The main evidence for that is the elasticity of the substitution of capital for labor in the output function, which is greater than one. Moreover, the observed trend toward increasing inequality r>g in Europe and the US is very likely to spread worldwide by 2050, with a difference of no less than 2.5%, between r and g, powering the mechanism responsible for the disparity between wealthy heirs and salaried or wage-earning workers, regardless of the latters' talents (p. 221): -on the basis of [Texte] 15/04/2014 historical data, one can estimate an elasticity [] between 1.3 and 1.6 […] the most likely outcome is that the volume effect will outweigh the price effect, which means that the accumulation effect will outweigh the decrease in the return on capital.‖ Piketty shows, in the end, that the relentless dynamic toward economic inequality, which operated at full throttle during the 19th century and was temporarily interrupted by the socio-economic upheavals of the 20th century, began a resurgence in the past three decades, virtually without notice. The share of labor in national income began lowering whereas the ratio of capital to income is likely to stabilize at very high levels. (Capital, which is mostly privately held, could represent up to 9 to 10 years worth of income by 2050.) That suggests that we and our children are very likely to live in world where the weight of the past bears down heavily on the economy as well as on everyone's social status. It is a world where inherited wealth dominates wealth earned by dint of work and talent, a world where a handful of wealthy heirs have ascendency over public powers and elected politicians. It is a world, too, where individual merit no longer gives hope of the greatest financial rewards, and where rich heirs and ordinary citizens live poles apart, ushering in a sweeping transformation of our civilizations and democratic regimes, on a scale heretofore unprecedented (p. 26):

These findings in Piketty's work speak to an economic worldview that demands comparison, most notably, with the Marxist cosmology.

A Piketty worldview: between Fukuyama and Rosanvallon

For the generation of Western Europeans educated during the post-war period, it was obvious that the surge in growth and the abatement of inequality they witnessed were not temporary but were, instead, deep trends ushering in a new world. Many pundits, particularly leftwing scholars, engaged in eschatological thinking where most inequalities, along with bourgeois contempt and arrogance, would eventually vanish. Piketty's cold analysis suggests the contrary: -In fact, when viewed in historical perspective, the thirty postwar years were the exceptional period‖ (p. 96) and they should never come back because -there is no historical example of a country at the world technological frontier whose growth in per capita output exceeded 1.5 percent over a lengthy period of time.‖

It is almost unheard of to come across an X-generation leftwing scholar who claims that the post-war years were an exception that very likely never will be repeated. Unlike the 20th century, the new era is set to be one of great stability, favoring the accumulation and concentration of capital, where output per capita will be at virtually the same level worldwide: -Per capita output in China, Eastern Europe, South America, North Africa, and the Middle East would match that of the wealthiest countries by 2050. After that, the distribution of global output […] would approximate the distribution of the population.‖ (p. 100)

A Left-wing Fukuyama

In certain respects, Piketty's worldview closely resembles the End of History foreseen by Francis [START_REF] Fukuyama | The End of History?[END_REF][START_REF] Fukuyama | The End of History and the Last Man[END_REF] immediately after the fall of Soviet Communism. No external upheaval is expected to destroy the stock of accumulated capital or prevent the global convergence of output. But unlike the centrist Fukuyama, the left-leaning Piketty does not [Texte] 15/04/2014 posit a positive, liberal eschatological view. Instead, he foretells of an era analogous with the 19th century fuelled by -grand dynamics‖ driving the accumulation of capital described above, and bringing about the sociopolitical disparity born of the extreme concentration of capital. An even global distribution of output per capita does not mean that income will be accumulated evenly or equitably. Rather, it is very likely that the income generated by billions of people will be captured by great wealth holders and transferred to remote regions, further widening imbalances, much to the discontent and dismay of the -serf‖ populations.

Rosanvallon's economic alter ego

But Piketty is much more than merely a left-wing Fukuyama. He is also a theorist of commonality (Rosanvallon, 2011, p. 382), which means he embraces the Enlightenmentinspired ideal of equality, whereby citizens in modern democracies live in a common world, regardless of any prevailing economic inequalities. More importantly, Piketty's Capital shares intellectual parallels with Rosanvallon's work on democracy and equality. Not only was Rosanvallon the director of the publishing imprint under which the original French edition of Capital in 21st Century was released (2013), Piketty proposed a major tax reform (2011) in another collection under Rosanvallon's aegis. More significantly, both scholars' worldviews complement each another perfectly. In short, Piketty's text can be seen as a kind of translation of Rosanvallon's ideas into an economic framework.

The dystopic side of that worldview is clearly exemplified, in the eyes of both scholars, by the world of the 19th century. Rosanvallon's argument, roughly summed up, is that 18th-century philosophers, in France, Scotland and America, built their concept of equality while totally unaware of the radical disparities, distancing the rich from the poor, that would be accentuated by the Industrial Revolution of the 19th Century. Those disparities, epitomized in Disraeli's famous novel Sybil, or The Two Nations, sparked anti-democratic ideologies as well as major upheavals in the following century. It is noteworthy that Piketty uses not only the novels of Jane Austen, and Balzac but also that of Disraeli (as does Rosanvallon) to situate his dystopia (a society dominated by holders of inherited wealth) but also views the 19th century as an era of economic regression compared with the Revolutionary and Empire periods: -When the lower classes of the Restoration and July Monarchy looked back on the revolutionary period and the Napoleonic era, they accordingly remembered good times‖ (p. 225). Piketty and Rosanvallon also reserve the same harsh criticism for -meritocratic extremism‖ [START_REF] Piketty | Le Capital au XXIe Siècle[END_REF], p. 334). On the one hand, Rosanvallon denounces the -principle of equality in chances restricted to the top of the society‖ described by Guizot, for instance, which was the mask of the low degree of intergenerational elasticity in terms of occupational rank or position society-wide [START_REF] Rosanvallon | La société des égaux[END_REF], p.126). On the other hand, Piketty offers the same type of argument regarding the illusory legitimacy of the extravagant compensation claimed by high-level executives [START_REF] Piketty | Le Capital au XXIe Siècle[END_REF], p. 334). Both scholars discount the legitimacy of the -super wages‖ of -super-managers,‖ with the same vehemence, but each advances his own line of argument. Piketty focuses on incentive systems. He describes the rise of super salaries as proof of the failure of so-called -corporate governance,‖ which results in a -pay for luck‖ system (pp. 334-335). Piketty even argues that the -hands in the till‖ metaphor (p. 332) is more apropos than the -invisible hand‖ to describe how executives go about capturing the firm's income. Rosanvallon, for his part, puts forward a similar argument but expressed in the politico-moral terms befitting his concern for political legitimacy: -These are neither virtue, nor merit, nor individual talent which determined the rise of highest wagesall the more than corporate failures hardly had impact on them. It is, in diverse manners, ruse, manipulation, bargaining, complicity, and even corruption that played the main role‖ (2011, pp. 331-332).

[Texte] 15/04/2014 When turning to the Utopian side of their worldviews, both authors embrace somewhat similar philosophies. To lend renewed credence and substance to the ideal of equality, in the face of the particular challenges of the 21st century, Piketty proposes a view that takes into account the forces that underlie economic disparity, principally by alluding, directly or indirectly, to three key concepts that Rosanvallon (2011) calls -singularity, reciprocity and commonality"2 (p. 22). Of these three, commonality is given pride of place in Capital. Piketty's main fear is the inexorably rising disparity between the poor and those who live off inherited fortunes (the first percentile), given that wealth inequality destroys commonality and, in so doing, undermines citizenship and democracy. Reciprocity is a second concept that Piketty addresses, if only tangentially but no less pointedly, when he offers the outlines a social Utopia: -We are free to imagine an ideal society in which all other tasks are almost totally automated and each individual has as much freedom as possible to pursue the goods of education, culture, and health for the benefit of herself and others. Everyone would be by turn teacher or student, writer or reader, actor or spectator, doctor or patient.‖ (p. 308) Finally, Piketty's and Rosanvallon's key argument consists in affirming that, -inequality, and that is the key point, does not have an impact solely on the deprived; it has deleterious effects on everyone‖ (Rosanvallon, 2011, p. 353).

Piketty's epistemology: against scholarly obscurantism as well Marxist obscurity

The relevance of Piketty's discussion on wealth stems from the originality and rigor of his method, which allow him to bridge an oft-ignored cross-disciplinary gap: -The historical and statistical study of tax records falls into a sort of academic no-man's-land, too historical for economists and too econometric for historians.‖ (p. 39).

Moreover, his epistemological approach recalls what scholars dedicated to the practice of management commonly experience when confronted with the arrogance of the advocates of so-called mainstream research. Mutatis mutandis, Piketty's reflections about the plagues of his discipline take on full resonance for business historians and management science academics.

Against scholarly obscurantism

How right and timely for Piketty to point out that the use and abuse of quantitative modeling in economics is long overdue for serious consideration, especially as this mature scholarly discipline has come to cast a long shadow over younger, though intimately related fields (such as management science), too often relegated to the footnotes of historical research: -The discipline of economics has yet to get over its childish passion for mathematics and for purely theoretical and often highly ideological speculation, at the expense of historical research and collaboration with the other social sciences […] this obsession with mathematics is an easy way of acquiring the appearance of scientificity without having to answer the far more complex questions posed by the world we live in.‖ (p. 32)

Another target of Piketty's opprobrium is -the profession's undue enthusiasm for simplistic mathematical models based on so-called representative agents theories‖ (p. 38)

[Texte] 15/04/2014 (Agency theory being typically one of them), which leads to it totally overlooking the reality of social differences and diversity, while also ignoring major problems. He calls into question not only the use and misuse of dis-embedded theories imported directly from economics into management science, but also warns management science scholars against general reliance on quantitative models of psychology, whose quantitative tests of construct validity are often taken as the Alpha and Omega (or rather, the alpha and the rho) of every rigorous study. Classical quantitative psychology is based on the assumption that type psychology is a legitimate way of predicting universal characteristics of behavior within an individual. While these methods may have a certain usefulness for management science studies underpinned by universal classifications of different types of personalities, aren't management science scholars who apply such models to predict collective behavior (within organizations, for example) guilty of the same mistakes committed by economists who rely on principal-agent theory? After all, doesn't this amount to casting aside social reality in favor of a puerile fascination for statistical tools? What scientific rigor can be shown in treating organizations as if they were individuals that can somehow be deciphered using the tools of quantitative psychology? Indeed, Piketty's epistemological affirmations should serve as an indictment of scholarly obscurantism in Human Sciences3 , an obscurantism that has long manifested itself in studies on -absorptive capacity‖ or -dynamic capability‖ where organizations are attributed human types or qualities, for no real scientific purpose.

To transition to the question of the alleged Marxist influences in Piketty's work, let us now consider the harsh criticism he has leveled at the economic indicators used by institutional publications, including the OECD, to measure inequality. Piketty argues, in particular, that, because economists usually do not break down figures so far as to look at the wealthiest individuals with a personal income at the 10 th percentile (or 1 st decile), the true breadth of the current inequality gap has been underestimated: -The official reports of national and international agencies are supposed to inform public debate about the distribution of income and wealth, but in practice they often give an artificially rosy picture of inequality‖ (p. 268). Piketty lays particular blame on the famous Gini index, an internationally accepted measure of income inequality, which, he contends, obscures the reality of the extreme concentration of wealth within the upper-most bracket of the economic elite. The result: it provides -an abstract and sterile view of inequality, which makes it difficult for people to grasp their position in the contemporary hierarchy‖ (p. 267) and, consequently, contributes to dissimulating the emergence of a society of inherited wealth holders.

A Marxist gesture: the will to demystify

In his attempt to pull back the -chaste veil‖ (p. 267) that conceals the reality of inequality and disparity, Piketty takes a path somewhat analogous to Marx's. And, in that regard at least, he can be spoken of in the same terms used by Marxists scholars to describe their great predecessor and path breaker. Like Marx, Piketty exhibits -a fierce determination to free oneself from the myths that were taken as truth, and his experience of the real history which has shaken up and swept away those myths‖ (Althusser, 2005(Althusser, [1965a]], p.80). Indeed, the will to reveal what economic scholarship had sought to make invisible should be acknowledged as a fully Marxist gesture. We may also say about Piketty's work what Althusser said in Reading Capital: -Understanding that paradoxical and necessary identity between seeing and not seeing in the sight itself, is exactly what defines our problem‖ (1996, [1965b], p. 14). Ultimately, however, there is one idea, and one alone, that justifies drawing an analogy between Piketty's and Marx's master works: the authors' will to demystify [Texte] 15/04/2014 economic models and statistics that have been taken for granted and gone virtually unquestioned--a feat that has played no small part in ensuring the acclaim and importance of their books. To be sure, Piketty acknowledges what he considers to be the major contribution of Marx: the -principle of infinite accumulation‖ of capital -that is, the inexorable tendency for capital to accumulate and become concentrated in ever fewer hands, with no natural limit to the process‖ (p.9). Nevertheless, two points mark a clear distinction between Piketty and Marx's thinking.

Not a Marxist, theoretically or epistemologically

The first point consists in the fact that Piketty clearly defines the contributions and limits of Marx's thoughts on his own methodology. Although Piketty acknowledges the importance of Marx's principle of infinite accumulation, he stresses that the Marxian apocalypse did not come to pass, simply because the increase in productivity and demographic growth offset the perpetual accumulation of capital over time (p. 228), as shown in the asymptotic law  = s/g above. Accordingly, Piketty recognizes what he deems to be Marx's genuine contribution, namely, the concept of the inexorable rise of capital, but nothing more than this (p. 234). He reserves his strongest criticism for Marxist methodology. Firstly, he points out the fragility of the Marxist empirical framework: -The problem is that despite these important intuitions, Marx usually adopted a fairly anecdotal and unsystematic approach to the available statistics‖ (p. 229). Secondly, there is some confusion and even obscurity in what Marx really might have meant to say: -Marx did not use mathematical models, and his prose was not always limpid, so it is difficult to be sure what he had in mind‖ (p. 228). These statements could have deeper implications for Marxists than for Marx himself, insofar as such words imply that millions of pages inspired by Das Kapital may be no more than fantastical imaginings borne out of ambiguity. It is no wonder, then, that [START_REF] Wallerstein | Le capitalism historique[END_REF] proposed to -take seriously the claim [Marx] was not Marxist,‖ whereas Althusser, the High Priest of French Marxism, admitted, by the end of his life, that -in a certain sense, Aron was right. We have fabricated an -imaginary‖ philosophy for Marx, a philosophy that did not exist in his work‖ (1994, quoted by Garo in Bourdin, 2008, p. 33). This means we must scrutinize not only the theoretical, but also the epistemological and philosophical gaps that separate Piketty from Marxism.

The second point is that the path chosen by Piketty's diverges radically from that taken by genuine Marxists, at least when viewed from two perspectives. Notably, there is the rather minor role that Piketty attributes to factors of production. A modern Marxist would focus on new conditions of production and, on that basis, paint the big picture of the world to come, along the example of Yann Moulier-Boutang's -cognitive capitalism‖ (2007). In contrast, Piketty focuses essentially on property whether in its classical form (land and income producing property) or the modern-day equivalent called housing wealth (building and housing values). Property ownership is for Piketty as important as the ownership of productive capital in fueling the dynamics of inequality and the -great disparity.‖ The line of demarcation between Piketty and Marxism is fixed by the definition and role attributed to ideology. Piketty wants to pull back the veil of ignorance that conceals the real mechanism regulating the disparity that separates the wealthy at the first percentiles of income brackets from the rest of the population. His goal, in so doing, is to initiate a broad debate, and he honestly believes that an enlightening political discussion can influence the future. Dyed-inthe-Wool Marxists, on the other hand, believe(d) that all such debates only scratch the surface of deep-rooted societal problems. Only a revolutionary upheaval of the economic infrastructures (during the preliminary phase of dictatorship of the proletariat) can make a difference. What is more, the -thought of suspicion‖ advocated by Marxists is so radical that, in their view, every thought belongs to ideology. For Althusser, the concept of ideology is very general: -Ideology is a system (with its own logic and rigour) of representation (images, myths, ideas or concepts, depending on the case) endowed with a historical existence and role within a given society‖ (2005, [1965a], p. 238). That generality engenders a view in which ideology is so pervasive that no one can escape it, for -ideology is not an aberration or a contingent excrescence of History: it is a structure essential to the historical life of societies‖ (ibid, p. 239), even for scholars in any forthcoming communist society. -Historical materialism cannot even conceive something like a communist society able to give up ideology‖ (ibid., p. 239). As we can clearly see, Piketty does not share that same view of the roles of scholarship, thought and debate in society. Unlike the thinkers of radical suspicion, he believes in reason, which enroots his approach far more firmly in Enlightenment or Kantian belief in reason, within limits, than in Marxism or any kind of Marxist philosophy.

Even though Piketty addresses issues that greatly overlap with Marxist considerations, he defines and analyzes problems in a quite different manner. His cross-disciplinary vision (p. 64) is a somewhat scholarly answer to Rosanvallon's call for new approaches: -The challenge is to build a ‗broadened political economy' of social relationship, making it possible to ground a genuine theory of equality that would incorporate its various dimensions in order to provide reforming action with solid and universal bases‖ (2011, p. 354). In opposition to Marxian obscurity and Bolshevist darkness, he proposes the search for enlightenment through reason.

Conclusion: towards a new Enlightenment … or darkness?

In the end, Piketty's Capital in the Twenty-First Century is more than just merely an answer to the key issues raised 30 years earlier by Braudel's illustrious disciple, Immanuel Wallerstein, in his series of conferences titled Historical Capitalism (1983). Wallerstein proposed a definition of historical capitalism that was squarely focused on the dynamic of capital accumulation, much like Piketty. Speaking of historical capitalism, he says, -It is that social system in which the scope of these rules (the law of value) has grown even wider, the enforcers of these rules ever more intransigent, the penetration of these rules into the social fabric ever greater, while social opposition to these rules has grown ever louder and more organized.‖ (p. 17) 4 In 1983, Wallerstein made the following prophesy: -Historical systems are decidedly historical. They are born and ultimately come to an end […]. Historical capitalism entered into a structural crisis in the early 20th century and will certainly end up as a historical system during the subsequent one‖ (pp. 92-93). In parallel, Wallerstein had identified a major disparity between wealthy countries and developing economies, noting, -a widening gap between the income earned by the happy and wealthy few, who make up 10% to 15% of the worldwide population, and the income earned by the rest of the world‖ (p. 105). He concluded by giving a somewhat vague, gloomy outlook: -The current system cannot continue to exist; it is losing its capacity for self-regulation. But the exact outcome in the new world-system(s) cannot be predicted‖ (p. 128).

Not only does Piketty predict great geo-economic shifts, he also presents an outlook that constitutes a radical reversal in the long-run trajectory of capitalism described by Wallerstein. Whereas the latter looks to the events of the 20th century to diagnose a crisis that will ultimately bring about the end of capitalism, the former explains that it is precisely the [Texte] 15/04/2014 century's world wars that rejuvenated capitalism and delayed the widening disparity between the capital holders and the rest of the population. In certain respects, Piketty and Wallerstein are in agreement, insofar as they see critical bifurcations in the dynamic of capitalism in the 21st Century. But on another score, their conclusions are somewhat divergent. According to Piketty, democracy and citizenship are in jeopardy, not capital, which continues on its drive toward accumulation and concentration, more relentlessly than ever before. Piketty's view on concentration is even quite hyperbolic, with his measure of the actual weight of the 1% and 0.1% wealthiest, compared with the wide 10-15% happiest focused on by Wallerstein. Let us point out one, last curious analogy between Piketty's work and Marxist thinking, which is especially revelatory of their differences. Feuerbach (1841) wanted to carry out a reversal of Hegel's dialectics and Marx reiterated a reversal of that reversal in his Theses on Feuerbach (Macherey, 2008). Here, we can observe that Piketty carries out a reversal of the -Story of Capitalism‖ hitherto narrated by Braudel (one of his great influences) and his disciple Wallerstein: the 21st century may not witness the extinction of capitalism, but it may well serve as the backdrop for an even more merciless world-system of capital accumulation and concentration and-conversely-of submission and deprivation. All of which leads us to a dystopic reversal of that reversal: capitalism may indeed become extinct, as Wallerstein predicted, and the first five-hundred years of its development may be viewed as merely a rehearsal for a still deeper, crueler concentration and oppression, in a world where no geographical frontiers or voice of civilization rises up to challenge the domination of the wealthiest capital holders.

Quotations from Piketty's book are drawn directly from the English translation

Book Review Piketty

Author's translation of -singularité, réciprocité et communalité‖.

Significantly, the author addresses this point both in the introduction (p.32) and in the conclusion (p. 575) of his opus magnum.

Author's translation based on the French re-edition of 2011.

Acknowledgements

Many thanks to Randall Cherry who achieved the copy editing of that paper on the basis of the original English version.


J Joie

Y Lei

T Colin

M.-C Durrieu

C Poignard
email: clair.poignard@inria.fr

O Saut

Modelling of migration and orientation of endothelial cells on micropatterned polymers

Keywords: Tissue engineering, biomathematical modeling, agent-based model, endothelial cells, migration, differential equations Ingénierie tissulaire, modélisation bio-mathématique, modèles d'agents, cellules endothéliales, migration, équation différentielles

Understanding the cell migration on micropatterned polymers, as well as their orientation is a critical issue in tissue engineering, since it is the preliminary step in the blood vessel formation. In this paper, we derive an agent-based model to describe the migration and the orientation of endothelial cells seeded on bioactive micropatterned polymers. The model is obtained thanks to the experimental observations, and it provides numerical results that are quantitatively in accordance with the experimental data.

Introduction

Tissue engineering is the use of combination of cells, engineering, materials, and suitable biochemical factors to improve or replace biological functions [START_REF] Nerem | Tissue engineering: the hope, the hype, and the future[END_REF]. A major problem for the creation of tissue in vitro is the lack of functional microvasculature networks. Indeed, the role of capillaries and blood vessels is to supply oxygen and nutrients to the cells, but also to remove waste products and CO 2 [START_REF] Jain | Engineering vascularized tissue[END_REF]. Without an efficient vessel network, the size of cultured tissues is limited by the weak diffusion of nutrients. The understanding of the mechanisms involved in the creation of new blood vessel is therefore crucial.

One strategy to develop clinical implants consists of appropriate utilizations of bioactive materials: bioactive materials may induce in vivo regenerative response at the site of damage, whereas when used in vitro, they can stimulate the tissue growth for subsequent implantation [START_REF] Anselme | The interaction of cells and bacteria with surfaces structured at the nanometre scale[END_REF], [START_REF] Lei | Geometrical microfeature cues for directing tubulogenesis of endothelial cells[END_REF]. Different bioactive ligands have been used to study their effects on cell functions for a better understanding of vascularization [START_REF] Phelps | Engineering more than a cell: vascularization strategies in tissue engineering[END_REF]. In the aim of promoting angiogenesis in the case of tissue engineering or of inhibiting angiogenesis in the case of cancer, it is important to understand the mechanisms that regulate lumen formation. Successful micropatterning of cells is becoming a key component of this field [START_REF] Ito | Surface micropatterning to regulate cell functions[END_REF].

Researchers are currently interested in the behavior of cells on substrates that have been patterned by micro-or nano-fabrication [START_REF] Folch | Microengineering of cellular interactions[END_REF][START_REF] Nicolau | Patterning neuronal and glia cells on light-assisted functionalized photoresists[END_REF]. As reported by Chen et al. in [START_REF] Chen | Geometric control of cell life and death[END_REF], cell positioning and physiology can be controlled by the substrate on which the cells adhere. Experiments presented by Lei et al. in [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF][START_REF] Lei | Geometrical microfeature cues for directing tubulogenesis of endothelial cells[END_REF] show that the use of cell adhesion peptides that are micropatterned onto material makes possible the formation of tube-like structures unlike the use of homogeneously grafted materials.

More precisely, experimental studies using micropatterned substrates revealed that the cell migration is governed by the geometry of patterns. So cultured endothelial cells form extensive cell-cell interactions, and in some configurations, the accumulation of endothelial cell junctions implies that tube-like structures are created. The goal of the present paper is to provide a simple model that describes quantitatively the cell migration towards the micropatterns and the final orientation of the cells on the strips.

In a previous work [START_REF] Colin | Modeling of the migration of endothelial cells on bioactive micropatterned polymers[END_REF], we have derived a continuous model of cell migration based on the well-known Patlak-Keller-Segel model for chemotaxy. That approach made possible qualitative description of the evolution of endothelial cell densities in accordance with the experiments. In particular we have shown the high influence of the geometry and of the initial cell density on the migration.

Nevertheless such a continuous model does not take the cell orientation into account, which is an important experimental data since it describes the cell organization on the micropatterns. Cell orientation on the micropatterns plays a crucial role in the formation of tube-like structures, and the aim of this work is to propose a simple model that can account for it.

Since the experiments deal with a quit small number of cells (around 50 000 per cm 2 ), unlike [START_REF] Colin | Modeling of the migration of endothelial cells on bioactive micropatterned polymers[END_REF], it is relevant to consider each cell separately and we describe cell-to-cell interactions for a large number of cells. We refer to [START_REF] Drasdo | Cell-based models of avascular tumor growth[END_REF][START_REF] Tsai | Simulations of endothelial cells clusters migration in angiogenesis[END_REF] for agent-based models that describe cell-cel interactions. We emphasize that the idea is to propose a very simple model that can be fitted to produce quantitative results in accordance with the data. We thus started from the biological observations of the phenomena, and we derive our models thanks to simple mechanics considerations.

Polarization of cells can be an active phenomena induced by external causes (see [START_REF] Hawkins | Rebuilding cytoskeleton roads: Active-transport-induced polarization of cells[END_REF], [START_REF] Madden | Cell polarity and morphogenesis in budding yeast[END_REF], [START_REF] Bagnat | Cell surface polarization during yeast mating[END_REF]). Nevertheless in some cases polarization can be observed without any external stimulations as reported in [START_REF] Irazoqui | Scaffold-mediated symmetry breaking by cdc42p[END_REF], [START_REF] Wedlich-Soldner | Robust cell polarity is a dynamic state established by coupling transport and gtpase signaling[END_REF] and [START_REF] Wedlich-Soldner | Spontaneous cell polarization through actomyosin-based delivery of the cdc42 gtpase[END_REF]. Therefore, we have chosen some ad hoc rules of cell alignment in order to model the cell orientation. These phenomenological and simple rules account for the complex phenomenon of cell orientation, which is probably a mixing of the effect of the micro Inria patterned and of non observable phenomena.

The outline of this article is the following: in the first part we present the experimental techniques and the data that are available. We then describe the set of equations defining the model, before presenting numerical results. We also show the consistency of our model, by comparing quantitatively both numerical and experimental data. It is very striking and interesting that our very simple model, which is only based on classical mechanics without any description of the complex molecular interactions, provides results that are quantitatively in accordance with the experiments.

Experimental studies

The experimental results have been precisely presented in [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF][START_REF] Lei | Geometrical microfeature cues for directing tubulogenesis of endothelial cells[END_REF]. We only remind the reader of the main observations given in these papers. The micropatterned polymers are adhesive areas composed of cell adhesion peptides or growth factors peptides that make the cells adhere. These areas are surrounded by non-adhesive areas on which cells move freely [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]. Human umbilical vein endothelial cells (HUVECs) were isolated from the human umbilical cord vein. HUVECs were grown on gelatin coated culture flasks in a complete HUVEC culture medium. HUVECs are seeded onto micropatterned bioactive materials during several hours, then they are washed out. Only the adhered endothelial cells remain on material. The initial cell density is around 50 000 cells per cm 2 (see [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]).

Active principles (cell adhesion peptides or growth factors) do not diffuse in the domain. Therefore endothelial cells outside adhesive areas have no means to "feel" directly these areas. They find out these areas indirectly, by the chemoattractant produced by the cell that are already on the strip. Actually, during the migration phase, we observe that cells have a random motility and stop on adhesive areas. Moreover the attraction of endothelial cells on adhesive areas seems to be higher than the one of cells outside those areas. Note that throughout the paper we do not consider the influence of nutrients and we assume that cells obtain enough nutrients from the material (due to grafted active principles onto material).

In Fig. 1, the final position of the cells on two different micropatterns are observed. Fig. 1(a) shows the case of thin adhesive areas of 10 µm width with an inter-pattern separation of 100 µm, whereas Fig. 1(b) shows the end of the migration on large strips (bioactive pattern size: 300 µm and distance between patterns: 100 µm). For a detailed description of such experiments, one may refer to [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]. [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]. The distance between bioactive patterns is 100 µm.

It has been observed that cells tend to gather along the micropatterns. Note that the cells located on the adhesive areas still move along the strip: they are not completely stuck on the adhesive area, but they do not leave the strips. Note also that according to pattern size, endothelial cells line their cytoskeleton to adjust it with the adhesive area.

As presented in [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF], the orientation of the endothelial cells is strongly influenced by the width of the patterns. Fig. 2 shows cells on different widths of strips . One can also notice that, on micropatterns of 10 µm thin strips, tubes containing a central lumen may appear [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF], [START_REF] Dike | Geometric control of switching between growth, apoptosis, and differentiation during angiogenesis using micropatterned substrates[END_REF]. In other words, blood vessels are created from an initial random density of endothelial cells. Such phenomenon is not observed with larger strips [START_REF] Dike | Geometric control of switching between growth, apoptosis, and differentiation during angiogenesis using micropatterned substrates[END_REF], [START_REF] Kamei | Endothelial tubes assemble from intracellular vacuoles in vivo[END_REF], [START_REF] Lubarsky | Tube morphogenesis: making and shaping biological tubes[END_REF].

Our goal is to predict quantitative information on the influence of the micropatterned geometry on both cell migration and alignment. The following facts have been observed experimentally:

• Initially, cells have a random motility and stop on the adhesive areas.

• The attractive force between cells is important.

• Cells on adhesive areas more strongly pull their neighbors and still have a very low motility.

• Two cells cannot stack themselves.

• If part of a cell body is located on a strip, it tries to completely enter the strip.

• Cells tend to align themselves with their neighbors.

We aim at deriving a model that describes precisely these phenomena. Note that since the cell deformation during the migration is a very complex process, which is still poorly understood, we choose to consider a simple cell shape, namely in the paper, the cells are ellipses whose parameters depend on the configuration of the experiments.

Inria 3 Description of the model

Since the experimental data deal with a quite small number of cells, the continuous model derived in [START_REF] Colin | Modeling of the migration of endothelial cells on bioactive micropatterned polymers[END_REF] cannot provide results that could be quantitatively compared to the experiments. The use of a discrete model has been preferred here to describe the cell migration on the micropatterns at the microscopic level. We therefore aim at deriving a model that describes both motility and orientation of a single cell, with respect to its neighbors and its environment.

For the sake of simplicity, we did not consider the cell deformation caused by its motion. In addition, since the cells can be seen as flat 1 and since in most of our data they do not overlap, we consider cells to be bidimensional.

We denote by Ω the domain on which cells migrate. This domain is split into two parts: the micropatterned domain Ω π , on which cells adhere, and the domain Ω \ Ω π , on which cells move freely. In the experiments, Ω π is mainly composed of disjoint thin strips of polymers, but some experiments have dealt with annulus micropatterns (see Fig. 3(c)). We describe the motion and the orientation of N cells on the domain Ω.

For 0 ≤ i ≤ N , the i th cell at the time t is characterized by an ellipse centered at the point M i (t) of the surface and whose (major axis) orientation is given by the unit vector L i (t), as described in Fig. 3(a). The major and minor radii of the ellipse, respectively denoted by Λ and λ, do not depend on time: they are parameters of the model that will be fitted with the experimental data. The heuristics of the modeling can be summarized as follows. Using the principle of the second Newton's law, which states that the net force is proportional to the acceleration, we first obtain the motion of M i , the center of the ellipsoidal cell. We then obtain the equation on L i by analogy with the orientation of a dipole subjected to a field in a bidimensional domain.

Net force acting on the cell

According to the experiments, and to the biological consideration, a single cell is subjected to the four following forces:

• A long range attractive force between the cells F c , generated by the chemical signal released by the other cells, so that cells attract themselves.

• A short range repulsive force F r , which prevents the overcrowding of the cells.

• A friction force F f , which describes the adhesion to the domain.

• An attractive force on the strip F a , which stacks the cell on the strip thanks to the nutrients.

The differential equation satisfied by the cell center M i reads:

M i = F c -F r -F f + F a . (1)
Note that we have normalized cell mass to 1 in Newton's law. This is hidden in the parameters describing each force. We chose to keep the inertial term Mi because it is not clear if it can be neglected in the crowding process. The drawbacks of this choice is that small time steps have to be used, which increase the computational time. Of course one can recover the usual overdamped model by choosing carefully the coefficient of the forces which neglect the inertial term. Moreover, as our model involves the velocity of the cells in the description of the forces, it would be difficult to solve numerically the problem without taking the acceleration into account.

Let us write precisely the above forces.

Cell-to-cell attraction force F c

The way that endothelial cells communicate is still not clearly understood, but according to biologists (see [START_REF] Lei | Biochemical and microscale modification of polymer for endothelial cell angiogenesis[END_REF]), it seems that cells produce chemoattractant that diffuse around them in order to attract their neighbors. We denote by ϕ the density of chemoattractant produced by the cells on the domain Ω. For each cell, the production of ϕ is generated by a Gaussian Ψ supported by the cell, whose maximum is at the cell center. We assume that the diffusion of ϕ occurs on a smaller timescale than the cell movement. Therefore ϕ satisfies the Poisson equation:

∆ϕ = f c (x) N i=1 Ψ (x -M i (t, x)) , ∀(t, x) ∈ (0, ∞) × Ω.
(

) 2
with homogenous Dirichlet boundary conditions.

Here above the function f c is related to the rate of production of the chemoattractant. We assume that cells produce more chemoattractant on the strip than outside, therefore f c is the following piecewise constant function:

f c (x) = 1 if x ∈ Ω \ Ω π γ c if x ∈ Ω π , with γ c > 1. (3)
This corresponds to the fact that cells that are in good condition of adhesion tends to attract the other cells more than the cell whose adhesion is weak. The attraction force F c is then derived from ϕ:

F c (t, x) = ∇ϕ(t, x), ∀(t, x) ∈ (0, +∞) × Ω.
From the numerical point of view, ϕ (and thus the attraction force F c ) is computed on the cartesian grid, while the cells move freely in the domain. Therefore, the value of F c is projected on each endothelial cell (M j ) N j=1 , using the projection method described by Min and Gibou [START_REF] Min | A second order accurate level set method on non-graded adaptive cartesian grids[END_REF].

Inria

Repulsive force

It has been experimentally observed that two cells do not overlap. This fact is taken into account by a repulsive term which prevents the cell overcrowding:

R(M i ) = N j=1 j =i M i -M j M i -M j 3 . ( 4
)
The power 3 ensures that the repulsive force is an inverse-square law: it is inversely proportional to the square of the distance between 2 cells. We chose such an inverse square law so that repulsion force is bigger than the attraction force. Actually in 2 dimensions the Green function of the Laplacian is log(r) and therefore the attraction force is of order 1/r: to prevent the overlapping, we have to impose a repulsion greater than the attraction for short distances.

Since the cells are ellipsoidal, it is necessary to add a multiplication factor in front of R. Actually suppose that two cells are located at the points M i and M j (see Fig 4). When cells are aligned along -----→ M i M j ⊥ (see Fig. 4(a)), the repulsion force is the lowest since the distance between the cell boundaries is the greatest. On the contrary, in the configuration of Figure 4(b) the cell alignment is parallel to -----→ M i M j , the repulsion force reach its maximum value. We therefore introduce the factor β(L i ), which depends on the orientation of the i th cell:

β(L i ) = Λ -λ λ N j=1 j =i M i -M j M i -M j • L i + M i -M j M i -M j • L j + 1. (5)
The coefficient (Λ -λ)/λ is introduced in order to take the cell shape into account: for rounded cell it vanishes since there is no cell orientation, and the more elongated the cell, the greater it is. The repulsive force now writes

F r (L i , M i ) = γ r β(L i )R(M i ), (6)
where γ r is a parameter fitted with the experiments.

Friction force

The friction force is proportional to the cell velocity, it thus reads

F f (M i ) = α f f f (M i ) Ṁ i ,
where the friction coefficient f f describes the adherence of the domain. Since cells move slowly on the strip, f f (M i ) is a piecewise constant function given by :

f f (M i ) = 1 if M i , ∈ Ω \ Ω π γ f if M i , ∈ Ω π , with γ f > 1. (7)

Attractive force of the adhesive areas

If a part of a cell is located on the micropatterned polymer region, it will find nutrients and oxygens on the active principle that will attract the whole cell on the strip. On the contrary, if the cell is completely on the polymer region, or outside this region, the attraction force F a due to the strip vanishes. We use the above function f f , which describes the adherence of the domain, to determine whether the cell is straddling the polymer region and the outer domain. Actually, since the distance between two strips is larger than the cell size, the term

f f (M i + ΛL i ) -f f (M i -ΛL i )
vanishes if the point M i + ΛL i and M i -ΛL i are on the same domain. It is positive and equal to γ f -1 if M i + ΛL i is on the strip and not M i -ΛL i , and thus the cell will tend to move in the direction of L i . On the other hand, if M i -ΛL i is on the strip and not M i + ΛL i , then it equals 1 -γ f , and the cell will tend to move in the direction of -L i . Similar considerations by replacing Λ by λ and L i by L ⊥ i ensures that F a is thus given by

F a (M i , L i ) = γ a (f f (M i + ΛL i ) -f f (M i -ΛL i )) L i + γ a f f (M i + λL ⊥ i ) -f f (M i -λL ⊥ i ) L ⊥ i , (8)
where γ a is a numerical parameter.

Orientation of the cells

As mentioned before, the deformation of endothelial cells is a very complex process, which is not taken into account in our simple model. Nevertheless, the cell alignment plays an important role in the tube-like formation. We therefore have chosen to take this alignment behavior into account by modeling the orientation L i of each cell.

Heuristics of cell alignment

Before stating the equation satisfied by L i , we introduce the "alignment" vector field

Φ defined from R 2 × R 2 onto R 2 by Φ : R 2 × R 2 → R 2 (U, V) → (U • V)(U • V ⊥ )U ⊥ (9)

Inria

In order to understand the above vector field Φ, suppose that V is a given vector of R 2 . Let U be a time-dependent vector of R 2 and let θ be the angle between U and V in the left-handed orientation. The orientation of U along V is easily written by the following ordinary differential equation on θ:

θ = - 1 2 sin(2θ),
such that {0, π} are stable equilibrium points while {-π/2, π/2} are unstable. This differential equation can be written in terms of U thanks to Φ:

U = Φ(U, V).

Differential equation for L i

According to the experiments, the cell orientation is driven by three phenomena:

• Alignment along the cell velocity.

• Alignment along the orientation of their neighbors.

• Alignment along the local tangent vector S i of the boundary of micropatterns at the closest point of M i .

We therefore infer the differential equation for L i :

Li = ω v Φ(L i , Ṁ i Ṁ i ) + ω c N j=1 j =i Φ(L i , L j ) M i -M j 2 λ 2 + ω π β π (M i )Φ(L i , S i ). (10)
Here again it is natural to introduce an inverse-square law for the interaction between two cells. The above function β π is given by

β π (M i ) = 4f f (M i ) -f f (M i + ΛL i ) -f f (M i -ΛL i ) -f f (M i + λL i ) -f f (M i -λL i ), (11)
it ensures that if the cell is entirely on (or out of) the micropatterns, the cell does not try to align but if the cell is partly on the strip then it tends to align along the direction of the strip.

Here above, the parameters ω v , ω c and ω π are homogeneous to the inverse of the characteristic time of each alignment. They will be fitted with the experiments.

Simulations and experiments

In order to study the consistency of our model, we compare the simulation with the experimental results. We focus on experimental results described in [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]. The domain Ω is a square of 1mm 2 , on which 500 cells are randomly distributed, in accordance with the experiments. To compute the distribution of the chemoattractant ϕ given by ( 2), the domain Ω is meshed thanks to a cartesian grid of 100 × 100 elements. Three different widths of the adhesive areas are considered: 10, 50 and 100 µm, with the same spacing of 100 µm between the strips. We emphasize that in order to validate the model we first optimized the parameters for one specific experiment: here we present the case of strips of 100µm width, and then, taking the values of the parameters for this specific experiment and changing the geometry, we compare the numerical results to the experimental data.

Numerical method

The discretization of the time derivatives are made using a "leap-frog" scheme for the acceleration and a centered scheme for the velocity. Initially, cells are randomly distributed on the domain Ω.

The Laplace problem (2) is discretized using a centered finite volume scheme (cf [START_REF] Eymard | Finite Volume Methods[END_REF]). The variables are computed at the center of each element of the mesh. The homogenous Neumann condition is imposed using a penalty method. This allows us to define a general framework to deal with different kinds of geometries (cf Fig. 3(b) and Fig. 3(c)). We use a solver implemented on the academic library eLYSe.

Calibration of the model

Cell shape

In [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF], for all three cases, the authors give the cell shapes distribution at the end of the experiments. They measured an approximation of the circularity C (cf Fig 5(a)) of the cells corresponding to the rapport of the major axis with the minor axis. They also give the area of the cells A on the three different geometries (cf Fig. 5(b)). These measures determine entirely the cell shape parameter of the model thanks to the relations:

C = λ/Λ, A = πΛλ.
Therefore, for each test-case the value of Λ and λ are given by table We first consider the experiments with strips of 100 µm width. We then chose the parameters given by table 2 to fit with the experiments. In the next paragraph, the consistency of the model will be shown in the next section by comparing the numerical results with the experiments. We keep the parameters of table 2 and we change the width of the strips as well as the final shape of the cells in accordance with the experiments.

γ c γ r γ f α f γ a ω v ω c ω π 10
A trickier question would be to solve an inverse problem and to find all the set of parameter that lead to an adequate fit of the data. Note that this is a difficult problem which we will not address in this paper. The above set of parameter fit with all the considered test cases.

Qualitative results on the cell motility towards the micropatterns

We first provide qualitative results of the cell migration for strips of 100 µm width. In Fig. 6, the positions of the endothelial cells at different times are given. The adhesive areas are colored in red, the outer domain in blue and the cells are plotted as white ellipses. In accordance with the experiments, we observe that the cells migrate towards the strips. Note that, as for the experiments, some cells remains out of the strips since they do not have enough space on the adhesive areas.

In the experiments the cells are washed out before counting. Numerically speaking, this corresponds to take out the cells that are not on the strips. Thus in Fig. 7, we present the final cell position on the strip only, at the end of the simulation (i.e. when the cells are almost completely stabilized).

The results are qualitatively in agreement with the experiments.

Quantitative results on cell alignment

We now focus on the orientation of the cells in the three previous configurations. In [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF], at the end of the experiment, the authors measure the angle θ (in degree) between the major axis of a cell with the axis of the adhesive areas as shown in the Fig. 8. For each configuration, six experiments are performed and histograms of the mean values are obtained. This is consistent with the number of experiments performed for each configurations ( [START_REF] Lei | Biochemical and microscale modification of polymer for endothelial cell angiogenesis[END_REF]). When θ is less than 10 • , we have considered that the cells are aligned with the strip. We first consider the 100 µm width strips case. We have computed the angle θ of the adhered cells for different initial conditions. We present experimental and numerical results in Fig. 9. The results are qualitatively and quantitatively in accordance with the experiments. Experimentally, 27% of the cells are considered to be aligned, numerically, this value is equal to 33.78%. Moreover, the mean value of the cell body alignment angle is equal to 25.9 in [START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF] and we obtain a closed value: 23.10. (see also Tables 34).

For the domain composed by 10µm width strips, the results are also in accordance with the experiments, as shown in Fig. 10.

Note that the number of aligned cells is more important on thin strips. Numerically, 70.54 % of the cells have an angle belonging between 0 • and 10 • whereas experimentally one obtains Let us now take 50 µm width strips. As previously, we show, in Fig. 11, the cell alignment at the end of the experiment.

The distribution of the angle have qualitatively the expected behavior, the cells are more aligned than in the case of 10 µm strips. One observes that 42% of the cells are aligned, when we have obtained 47.19%. We obtain an angle mean value of 14.62 • , which is very closed to the experimental measurements: 15.2 • .

Sensitivity analysis

We are interested here in studying the influence of the parameter involved on the orientation of the cells: ω v , ω c , and ω π on two different results: the mean angle value and the percentage of aligned cells. We use Morris's method introduced in [START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF]. We let one parameter vary and compute the elementary variation of the results for sample points of the parameter space. This

Conclusion

We have provided a simple agent-based model that describes the cell motility on bioactive micropatterned polymers thanks to classical mechanics consideration. The model has been derived by analogy to the Newton laws: the acceleration of the cell is driven by four forces, which describe Inria the cell-cell interactions and the influence of the domain on the cell motility. We also describe the orientation of the cell, similarly to a dipole subjected to a force field. This model involves phenomenological parameters whose values have been estimated to fit the experiments. Changing the configuration of the strips, but keeping the same parameters, we obtain results that are quantitatively in accordance with the experiments. The comparison between the experimental measurement of the cell orientation to the simulation are summarized in tables 3-4. As mentioned above, the numerical results are qualitatively in good agreement with the experimental data.

Percentage of aligned cells along the strips According to both experimental and numerical results (Fig. 10 11 and 9) the cell alignment is more efficient on thin strips than on larger ones.

Moreover, the forces that we have used are enough to describe both adhesion and orientation of the cells. This may be seen, in some sense, as a basic framework for the description of cell motion on scaffold.

Figure 1 :

 1 Figure 1: Endothelial cells alignment onto micropatterned polymer (PET) (10µm (a) and 300 µm (b) stripes of cell adhesion peptides)[START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]. The distance between bioactive patterns is 100 µm.

  (a) Without bioactive principle (b) Strip of 100 µm width (c) Strip of 50 µm width (d) Strip of 10 µm width

Figure 2 :

 2 Figure 2: Endothelial cells alignment on different micropatterned polymers (PET) (strips of RGDS peptides),[START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF]

  (a) Ideal cell (b) Micropatterns in strip (c) Micropatterns in circle

Figure 3 :

 3 Figure 3: Schematic single cell, and examples of experimental micropatterned domain.

Figure 4 :

 4 Figure 4: Influence of the cell orientation on the cell-cell distance.

  (a) Mean circularity C of the cells (b) Mean area A of the cells

Figure 5 :

 5 Figure 5: Cells shape informations measured at the end of the experiment. [15]

Figure 6 :

 6 Figure 6: Position and orientation of endothelial cells on 100 µm strips at different times.

2 Table 2 :

 22 10 -5 120 10 0.05 1 0.2 0.Parameters set to fit with the experiments of 100 µm.

Figure 7 :Figure 8 :

 78 Figure 7: Comparison between experimental results of [15] and numerical simulations

Figure 9 :

 9 Figure 9: Cell alignment on 100µm strips, comparison between experimental (a) and numerical (b) results.

Figure 10 :

 10 Figure 10: Cell alignment on 10 µm strips in the experiments Fig. 10(a) and in the simulation Fig.10(b).

Figure 11 :

 11 Figure 11: Cell alignment on 50µm strips, comparison between experimental (a) and numerical (b) results.

( a )

 a Influence on the mean angle value (b) Influence on the percentage of aligned cells

Figure 12 :

 12 Figure 12: Sensitivity analysis of the ω v (orientation along the velocity field), ω c (orientation along the neighbors of the cells), and ω π (orientation along the strips)parameters .

Table 1 :

 1 Major and minor axes of the cells with respect to the width of the strips. model does not come from physical consideration, the parameters involved in the modeling cannot be measured. Therefore it is essential to fit the parameters with the experiments.

	1. Since our phenomenological

Table 3 :

 3 Angle of the mean cell alignment along the patch Angle of the mean alignment along the strips 10 µm 50 µm 100 µm Exp. Data 8.6± 6.1 • 15.2±9.6 • 25.9±15.8 • Num. Res. 9.98 • 14.62 • 23.10 •

		10 µm 50 µm	100 µm
	Exp. Data	73	42	27
	Num. Res. 70.54	47.1	33.68

Table 4 :

 4 Cell alignment of the experiments (Exp. Data.) given by[START_REF] Lei | Modulation of lumen formation by microgeometrical bioactive cues and migration mode of actin machinery[END_REF] compared with the numerical simulations (Num. Res.)

RR n°8252

According to the experiments, the cells are

to 50 µm long, 10 to 20 µm wide and only 1 to 2 µm thick.RR n°8252

Inria

351, Cours de la Libération Bâtiment A 29

Publisher Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399