Numerical simulation of endocytosis - UCLAhelper.ipam.ucla.edu/publications/mms2014/mms2014_11862.pdf · Nicholas Michael Katz Princeton University 1966 Dirk Jan Struik Universiteit

Local inextensibility Membrane molecules Numerical results Summary

Numerical simulation of endocytosisDiffuse interface models for membranes with curvature-inducing

molecules

Sebastian Aland1,2 Jun Allard1 John Lowengrub (NYU PhD 1988) 2

1Technische Universitat Dresden, Germany2University of California Irvine, USA

International Workshop on Multiscale Modeling and Simulation, UCLA, April25, 2014

A 60th birthday celebration for Russ Caflisch.

Numerical simulation of endocytosis John Lowengrub, Dept Math/BME/ChEMS UCI


Mathematical genealogy

Jun AllardUniversity of British Columbia 2011

Vladimir Yurievich BaranovskyThe University of Chicago 2000

Long ChenThe Pennsylvania State University 2005

Michael CranstonUniversity of Minnesota-Minneapolis 1980

German EncisoRutgers University, New Brunswick 2005

Aleksandr FigotinTashkent State University 1980

Matthew Dean ForemanUniversity of California, Berkeley 1980

Anton GorodetskiMoscow State University 2001

Patrick Quirino GuidottiUniversität Zürich 1996

Hamid HezariThe Johns Hopkins University 2009

Svetlana Yakovlevna JitomirskayaMoscow State University 1991

Abel KleinMassachusetts Institute of Technology 1972

Natalia KomarovaUniversity of Arizona 1998

Katsiaryna KrupchykBelarusian State University 2003

Song-Ying LiUniversity of Pittsburgh 1992

John Samuel LowengrubNew York University 1988

Zhiqin LuNew York University 1997

Qing NieThe Ohio State University 1995

Karl Cooper RubinHarvard University 1981

Martin SchechterNew York University 1957

Richard Melvin SchoenStanford University 1977

Alice SilverbergPrinceton University 1984

Knut SolnaStanford University 1997

Jeffrey StreetsDuke University 2007

Chuu-Lian TerngBrandeis University 1976

Li-Sheng TsengThe University of Chicago 2003

Daqing WanUniversity of Washington 1991

Frederic Yui Ming WanMassachusetts Institute of Technology 1965

Liang XiaoMassachusetts Institute of Technology 2009

Jack Xue XinNew York University 1990

Yifeng YuUniversity of California, Berkeley 2005

Martin ZemanHumboldt-Universität zu Berlin 1998

Hongkai ZhaoUniversity of California, Los Angeles 1996

Eric CytrynbaumUniversity of Utah 2001

Victor A. GinzburgMoscow State University 1985

Jinchao XuCornell University 1989

Steven OreyCornell University 1954

Eduardo Daniel SontagUniversity of Florida 1976

Robert Martin SolovayThe University of Chicago 1964

Yulij S. IlyashenkoMoscow State University 1969

Herbert AmannAlbert-Ludwigs-Universität Freiburg im Breisgau 1965

Steven Morris ZelditchUniversity of California, Berkeley 1981

Yakov Grigor'evich SinaiMoscow State University 1960

Irving Ezra SegalYale University 1940

Alan Clive NewellMassachusetts Institute of Technology 1966

Nikolai Izobov

Frank Hayne Beatrous, Jr.Tulane University 1978

Russel Edward CaflischNew York University 1978

Gang TianHarvard University 1988

Gregory Richard BakerCalifornia Institute of Technology 1977

Saleh A. TanveerCalifornia Institute of Technology 1984

Andrew John WilesUniversity of Cambridge 1979

Louis NirenbergNew York University 1949

Lipman BersCharles University 1938

Leon M. SimonUniversity of Adelaide 1971

Shing-Tung YauUniversity of California, Berkeley 1971

Mark SternPrinceton University 1984

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George C. PapanicolaouNew York University 1969

Richard Sheldon PalaisHarvard University 1956

Jeffrey A. Harvey

Neal I. KoblitzPrinceton University 1974

M. Eric (Max) ReissnerMassachusetts Institute of Technology 1938

Kiran Sridhara KedlayaMassachusetts Institute of Technology 2000

Lawrence Craig EvansUniversity of California, Los Angeles 1975

Ronald Björn JensenRheinische Friedrich-Wilhelms-Universität Bonn 1964

Björn E. EngquistUppsala Universitet 1975

James Paul KeenerCalifornia Institute of Technology 1972

Alexandre Aleksandrovich KirillovMoscow State University 1962

James Henry BrambleUniversity of Maryland College Park 1958

John Barkley RosserPrinceton University 1934

Rudolf Emil KalmanColumbia University 1958

Saunders Mac LaneGeorg-August-Universität Göttingen 1934

Evgenii Mikhailovich Landis 1953

Joachim A. NitscheTechnische Universität Berlin 1951

Dietrich MorgensternTechnische Universität Berlin - Indiana University 1952, 1955

Alan David WeinsteinUniversity of California, Berkeley 1967

Andrei Nikolayevich KolmogorovMoscow State University 1925

Israil Moiseivich GelfandMoscow State University 1935

C. Einar (Carl) HilleStockholm University 1918

David John BenneyMassachusetts Institute of Technology 1959

Frank Thomas BirtelUniversity of Notre Dame 1960

William Robin ZameTulane University 1970

Philip G. SaffmanUniversity of Cambridge 1956

John Henry CoatesUniversity of Cambridge 1969

James J. StokerEidgenössische Technische Hochschule Zürich 1936

Charles LoewnerUniversity of Prague 1917

James H. MichaelUniversity of Adelaide 1957

Shiing-Shen ChernUniversität Hamburg 1936

H. Blaine (Herbert) Lawson, Jr.Stanford University 1969

Joseph Bishop KellerNew York University 1948

Herbert Bishop KellerNew York University 1954

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George Whitelaw MackeyHarvard University 1942

Pierre RamondSyracuse University 1969

Nicholas Michael KatzPrinceton University 1966

Dirk Jan StruikUniversiteit Leiden 1922

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Michael Grain CrandallUniversity of California, Berkeley 1965

Gisbert HasenjaegerWestfälische Wilhelms-Universität Münster 1950

Heinz-Otto KreissKungliga Tekniska Högskolan 1959

Lawrence Edward PayneIowa State University 1950

Alonzo ChurchPrinceton University 1927

John Ralph RagazziniColumbia University 1941

Hermann Claus Hugo WeylGeorg-August-Universität Göttingen 1908

I. Paul (Isaak) BernaysGeorg-August-Universität Göttingen 1912

Aleksandr Semenovich Kronrod 1949

Ivan Georgievich PetrovskyMoscow State University

Wolfgang Siegfried HaackFriedrich-Schiller-Universität Jena 1926

Werner Johannes SchmeidlerGeorg-August-Universität Göttingen 1917

Nikolai Nikolayevich LuzinMoscow State University 1915

Marcel RieszEötvös Loránd University 1912

C.-C. (Chia-Chiao) LinCalifornia Institute of Technology 1944

Irving Leonard GlicksbergUniversity of California, Los Angeles 1951

George Keith BatchelorUniversity of Cambridge 1948

Alan BakerUniversity of Cambridge 1964

Heinz HopfUniversität Berlin 1925

Hans FreudenthalUniversität Berlin 1930

George (György) PólyaEötvös Loránd University 1912

Georg Alexander PickUniversität Wien 1880

Wilhelm Johann Eugen BlaschkeUniversität Wien 1908

Robert OssermanHarvard University 1955

Richard CourantGeorg-August-Universität Göttingen 1910

Franz RellichGeorg-August-Universität Göttingen 1929

Marshall Harvey StoneHarvard University 1926

A. P. BalachandranUniversity of Madras 1962

Bernard Morris DworkColumbia University 1954

Willem van der WoudeRijksuniversiteit Groningen 1908

Jan Arnoldus SchoutenTechnische Universiteit Delft 1914

Frans OortUniversiteit Leiden 1961

Joseph H. M. SteenbrinkUniversiteit van Amsterdam 1974

Heinz Otto CordesGeorg-August-Universität Göttingen 1952

Heinrich ScholzFriedrich-Alexander-Universität Erlangen-Nürnberg 1913

Goeran BorgUppsala Universitet 1945

Wilhelm MagnusJohann Wolfgang Goethe-Universität Frankfurt am Main 1931

Dio Lewis HollThe University of Chicago 1925

Oswald VeblenThe University of Chicago 1903

John B. Russell, Jr.Massachusetts Institute of Technology 1950

David HilbertUniversität Königsberg 1885

Erhard SchmidtGeorg-August-Universität Göttingen 1905

Max W. DehnGeorg-August-Universität Göttingen 1900

Edmund LandauUniversität Berlin 1899

Dimitri Fedorowitsch EgorovMoscow State University 1901

Robert Karl Hermann HaußnerGeorg-August-Universität Göttingen 1888

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Theodore von KármánGeorg-August-Universität Göttingen 1908

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Lars Valerian AhlforsHelsingin Yliopisto 1932

George David BirkhoffThe University of Chicago 1907

Alladi RamakrishnanUniversity of Manchester 1950University of Manchester 1951

Emil ArtinUniversität Leipzig 1921

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Jacob CardinaalUniversiteit Utrecht 1903

Willem Titus van EstUniversiteit Utrecht 1950

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Arne BeurlingUppsala Universitet 1933

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Vannevar BushMassachusetts Institute of Technology 1916

C. L. Ferdinand (Carl Louis) LindemannFriedrich-Alexander-Universität Erlangen-Nürnberg 1873

Ferdinand Georg FrobeniusUniversität Berlin 1870

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Nicolai Vasilievich BugaevMoscow State University 1866

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Hermann Amandus SchwarzUniversität Berlin 1864

Ludwig PrandtlLudwig-Maximilians-Universität München 1899

Garrett BirkhoffUniversity of Cambridge

Joseph John ThomsonUniversity of Cambridge 1883

John Edensor LittlewoodUniversity of Cambridge 1907

C. Felix (Christian) KleinRheinische Friedrich-Wilhelms-Universität Bonn 1868

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Karl Theodor Wilhelm WeierstraßWestfälische Wilhelms-Universität Münster 1841

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Emil WeyrUniversity of Prague 1870

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Ernst Leonard LindelöfHelsingin Yliopisto 1893

Rolf Herman NevanlinnaHelsingin Yliopisto 1919

Maurice Stevenson BartlettUniversity of Cambridge

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Otto Ludwig HölderEberhard-Karls-Universität Tübingen 1882

David Bierens de HaanUniversiteit Leiden 1847

Jan de VriesUniversiteit van Amsterdam 1881

Nicolaas Govert de BruijnVrije Universiteit Amsterdam 1943

Kuno Ernst Berthold FischerMartin-Luther-Universität Halle-Wittenberg 1847

Anders WimanLund University 1892

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H. A. (Hubert Anson) NewtonYale University 1850

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Philip HallUniversity of Cambridge 1926

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Julius PlückerPhilipps-Universität Marburg 1823

Rudolf Otto Sigismund LipschitzUniversität Berlin 1853

Christoph GudermannGeorg-August-Universität Göttingen 1823

Humboldt-Universität zu Berlin 1832Heinrich Ferdinand Scherk

Universität Berlin 1823

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Johannes FrischaufUniversität Wien 1861

Karl FriesachUniversität Wien 1846

Robert Hjalmar MellinHelsingin Yliopisto 1882

John WishartUniversity College Cork

Hugo Hans von SeeligerUniversität Leipzig 1872

Ludwig BoltzmannUniversität Wien 1866

Gideon Janus VerdamUniversiteit Leiden 1825

Adrianus Jacobus van PeschUniversiteit Leiden 1890

Jurjen Ferdinand KoksmaRijksuniversiteit Groningen 1930

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Carl Fabian Emanuel BjörlingUppsala Universitet 1863

Michel ChaslesÉcole Polytechnique 1814

Simeon Denis PoissonÉcole Polytechnique 1800

Gustav Peter Lejeune DirichletRheinische Friedrich-Wilhelms-Universität Bonn 1827

Louis Jacques ThenardÉcole Polytechnique

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Karl PearsonUniversity of Cambridge 1879

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George Gabriel StokesUniversity of Cambridge 1841

Walter William Rouse BallUniversity of Cambridge 1874

Christian Ludwig GerlingGeorg-August-Universität Göttingen 1812

Martin OhmFriedrich-Alexander-Universität Erlangen-Nürnberg 1811

Bernhard Friedrich ThibautGeorg-August-Universität Göttingen 1796

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Jožef StefanUniversität Wien 1858

Jacob de Gelder

Simon Speijert van der EykUniversiteit Leiden

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Emanuel Gabriel BjörlingUppsala Universitet 1830

Joseph Louis Lagrange

Università di Torino 1754

Jean-Baptiste Joseph FourierÉcole Normale Supérieure

Pierre-Simon Laplace

Nicolas Louis Vauquelin

Walter Morley FletcherUniversity of Cambridge 1904

Sir Francis GaltonUniversity of Cambridge 1847

William HopkinsUniversity of Cambridge 1830

Isaac TodhunterUniversity of Cambridge 1848

Carl Friedrich GaußUniversität Helmstedt 1799

Johann Franz Friedrich EnckeUniversität Berlin 1825

Karl Christian von LangsdorfGeorg-August-Universität Göttingen / Justus-Liebig-Universität Gießen / Universität Erfurt 1781

Johann Salomo Christoph SchweiggerFriedrich-Alexander-Universität Erlangen-Nürnberg 1800

Abraham Gotthelf KästnerUniversität Leipzig 1739

Georg Christoph LichtenbergGeorg-August-Universität Göttingen 1765

Johann Friedrich PfaffGeorg-August-Universität Göttingen 1786

Johann Tobias MayerGeorg-August-Universität Göttingen 1773

Franz Josef Ritter von Gerstner 1777

Bernard(us) Placidus Johann Nepomuk BolzanoUniversity of Prague 1805

Göran DillnerUppsala Universitet 1863

Andreas von EttingshausenKaiserlich-königliche Technische Militärakademie / Universität Wien 1817

Jan Hendrik van SwindenUniversiteit Leiden 1766

Pieter NieuwlandAthenaeum Illustre Amsterdam 1783, 1789

Christiaan Hendrik DamenUniversiteit Utrecht 1783

Jan Cornelis KluyverPolytechnische School van Delft 1881

Rijksuniversiteit Groningen 1896

Friedrich Wilhem Joseph von SchellingEberhard-Karls-Universität Tübingen 1795

Jöns SvanbergUppsala Universitet 1796

Carl Johan MalmstenUppsala Universitet 1839

Leonhard EulerUniversität Basel 1726

Johann Friedrich HennertKönigliche Akademie der Wissenschaften zu Berlin

Giovanni Battista (Giambattista) Beccaria

Jean Le Rond d'Alembert

Antoine Francois de FourcroyUniversité de Paris

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Christian August HausenMartin-Luther-Universität Halle-Wittenberg 1713

Johann Elert BodeHandelsakademie Hamburg

Georg Friedrich HildebrandtGeorg-August-Universität Göttingen 1783

Franz August Wolf

Jan Tesánek 1748

Ignaz Lindner

Johannes LulofsUniversiteit Utrecht 1731Universiteit Utrecht 1734

Daniel Albert WyttenbachPhilipps-Universität Marburg 1760

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Johann Gottlieb FichteUniversität Königsberg 1792

Johann AfzeliusUppsala Universitet 1776

Johann BernoulliUniversität Basel 1690, 1694

Jean Baptiste Michel BucquetUniversité de Paris 1768

Micheal FosterUniversity of London 1859

Thomas JonesUniversity of Cambridge 1782

John Dawson

Johann Christoph WichmannshausenUniversität Leipzig 1685

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Johann Georg BüschGeorg-August-Universität Göttingen 1752

Johann Friedrich GmelinEberhard-Karls-Universität Tübingen 1768

Joseph Stepling 1743

Georg Jurij Bartolomej Veha von VegaLyceum of Ljubljana 1775

Jacob OdéUniversiteit Utrecht 1721Universiteit Utrecht 1727

Everhard OttoMartin-Luther-Universität Halle-Wittenberg 1714

Joseph Nicolas Delisle (de L'Isle )Académie royale des sciences de Paris 1714

Christian M. von WolffUniversität Leipzig 1703

Martin KnutzenUniversität Königsberg 1732

Samuel KlingenstiernaUppsala Universitet 1717

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August Gottlieb SpangenbergFriedrich-Schiller-Universität Jena 1726

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Immanuel KantUniversität Königsberg 1770

Torbern Olof BergmanUppsala Universitet 1758

Jacob BernoulliUniversität Basel 1676Universität Basel 1684

Nikolaus EglingerUniversität Basel 1660, 1661

Pierre Joseph MacquerUniversité de Paris 1742

Thomas Henry Huxley

William Sharpey 1823

Thomas PostlethwaiteUniversity of Cambridge 1756

John CrankeUniversity of Cambridge 1774

Edward WaringUniversity of Cambridge 1760

Henry Bracken

Otto MenckeUniversität Leipzig 1665

Johann PaschMartin-Luther-Universität Halle-Wittenberg 1683

Rudolf Jakob CamerariusEberhard-Karls-Universität Tübingen 1684, 1686

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Phillip Friedrich GmelinEberhard-Karls-Universität Tübingen 1742

Ferdinand Christoph OetingerMartin-Luther-Universität Halle-Wittenberg 1739

Ignatz Mühlwenzel

Gabriel GruberUniversität Graz 1767

Joseph Giuseppe Jakob von MaffeiUniversität Graz 1762

Joseph SerrurierUniversiteit Leiden 1690Universiteit Leiden 1690

Hieronymus Simons van AlphenUniversiteit Utrecht 1715

Nicolaus GundlingMartin-Luther-Universität Halle-Wittenberg 1703

Jacques CassiniUniversité de Paris 1691

Ehrenfried Walter von TschirnhausUniversiteit Leiden 1669, 1674

Gottfried Wilhelm LeibnizUniversität Leipzig 1666Universität Altdorf 1667

Académie royale des sciences de Paris 1676

Nicolas Malebranche 1672

Johann Franz BuddeusMartin-Luther-Universität Halle-Wittenberg 1687

Johann Matthias GesnerFriedrich-Schiller-Universität Jena 1715

Johann August BachUniversität Leipzig 1744

Johann August ErnestiUniversität Leipzig 1730

Christian Wilhelm KüstnerUniversität Leipzig 1742

Bengt FerrnerUppsala Universitet 1751

Pierre VarignonAcadémie royale des sciences de Paris 1687

Peter WerenfelsUniversität Basel 1649

Emmanuel StupanusUniversität Basel 1613

Johann Caspar BauhinUniversität Basel 1649

Georg Balthasar MetzgerFriedrich-Schiller-Universität Jena / Universität Basel 1644, 1650

Franciscus de le Boë SylviusUniversiteit Leiden / Universität Basel 1634, 1637

Gillaume-Francois RouelleUniversité de Paris 1725

Thomas Wharton JonesImperial College

Stephen WhissonUniversity of Cambridge 1742

Jakob ThomasiusUniversität Leipzig 1643

Christian ThomasiusUniversität Leipzig 1672

Europa-Universität Viadrina Frankfurt an der Oder 1679

Michael Walther, Jr.Martin-Luther-Universität Halle-Wittenberg 1661, 1687

Elias Rudolph Camerarius, Sr.Eberhard-Karls-Universität Tübingen 1663

Elias Rudolph Camerarius, Jr.Eberhard-Karls-Universität Tübingen 1691

Georg Erhard HambergerFriedrich-Schiller-Universität Jena 1721

Simon Paul HilscherFriedrich-Schiller-Universität Jena 1704

Burchard David MauchartEberhard-Karls-Universität Tübingen 1722

Michael AlbertiMartin-Luther-Universität Halle-Wittenberg 1703, 1704

Nikolaus Boda Poda von Neuhaus

Burchard de VolderUniversiteit Utrecht 1660Universiteit Leiden 1664

Herman BoerhaaveUniversiteit Leiden / Universiteit Harderwijk 1690, 1693

Paul HermannUniversità di Padova 1670

Melchior LeydeckerUniversiteit Leiden 1661

Johann Peter von LudewigMartin-Luther-Universität Halle-Wittenberg 1690

Jean-Dominique (Giovanni Domenico) Cassini

Arnold GeulincxUniversité Catholique de Louvain / Universiteit Leiden 1646, 1658

Rudolf Wilhelm KrauseUniversiteit Leiden 1671

Georg Wolffgang WedelFriedrich-Schiller-Universität Jena / Universiteit Leiden 1667

Pieter van SchootenUniversiteit Leiden 1655

Erhard WeigelUniversität Leipzig 1650

Bartholomäus Leonhard SchwendendörfferUniversität Leipzig 1656

Christiaan HuygensUniversiteit Leiden / Université d'Angers 1647, 1655

Johann Friedrich ChristMartin-Luther-Universität Halle-Wittenberg 1727

Mårten StrömerUppsala Universitet 1730

Theodor Zwinger, Jr.Universität Basel 1630

Petrus RyffUniversität Basel 1584

Johann Gottlob Spitzley

William McKenzieUniversität Wien 1818

Walter TaylorUniversity of Cambridge 1723

Friedrich LeibnizUniversität Leipzig 1622

Aegidius StrauchMartin-Luther-Universität Halle-Wittenberg 1651Martin-Luther-Universität Halle-Wittenberg 1657

Johann Andreas QuenstedtUniversität Helmstedt / Martin-Luther-Universität Halle-Wittenberg 1643, 1644

Johann Georg MacasiusFriedrich-Schiller-Universität Jena 1638, 1640

Johann Adolph WedelFriedrich-Schiller-Universität Jena 1694

Georg Ernst StahlFriedrich-Schiller-Universität Jena 1684

Johannes de BruynUniversiteit Utrecht 1644

Johannes HoornbeeckUniversiteit Utrecht 1643

Valentin AlbertiUniversität Leipzig 1678

Johann Rhetius

Erycius (Henrick van den Putte) PuteanusUniversität zu Köln / Université Catholique de Louvain 1595

Adolph VorstiusUniversiteit Leiden / Università di Padova 1619, 1622

Frans van Schooten, Jr.Universiteit Leiden 1635

Philipp MüllerUniversität Leipzig 1604

Jan Jansz Stampioen, Jr.

Karl Gottlieb KnorreMartin-Luther-Universität Halle-Wittenberg 1721

Anders Gabriel DuhreUppsala Universitet 1711

Sebastian BeckUniversität Basel 1610

Theodor ZwingerCollège de France 1553

Università di Padova 1559

John CraigUniversität Basel 1580

Nicolas Lemery

Georg Joseph BeerUniversität Wien 1786

Robert SmithUniversity of Cambridge 1715

Abraham Klein (Calovius)Universität Rostock 1632

Andreas Kunad

Georg CalixtUniversität Helmstedt 1607

Christoph NotnagelMartin-Luther-Universität Halle-Wittenberg 1630

Johannes MusaeusUniversität Erfurt 1634

Balthasar WidmarcterFriedrich-Schiller-Universität Jena 1640

Daniel BerckringerRijksuniversiteit Groningen 1640

Gisbertus VoetiusUniversiteit Leiden 1611

Martinus SchoockUniversiteit Utrecht 1636

Justus (Joost Lips) LipsiusUniversité Catholique de Louvain 1569

Gilbert JacchaeusUniversity of St. Andrews / Universität Helmstedt / Universiteit Leiden 1601/1603/1611

Franck Pieterszoon BurgersdijkUniversiteit Leiden 1614

Adriaan van den SpieghelUniversité Catholique de Louvain / Università di Padova 1603

Werner RolfinckMartin-Luther-Universität Halle-Wittenberg / Università di Padova 1625

Jacobus GoliusUniversiteit Leiden 1612, 1621

Marin MersenneUniversité Paris IV-Sorbonne 1611

Gilles Personne de Roberval

Christoph MeurerUniversität Leipzig 1582

Petrus (the Elder) ElviusUppsala Universitet 1688

Johann Jacob GrynaeusEberhard-Karls-Universität Tübingen 1564

Bassiano LandiUniversità di Padova 1542

Vittore TrincavelliUniversità di Padova

Petrus (Pierre de La Ramée) RamusCollège de Navarre 1536

Johannes (Johan van Heurne) HeurniusCollège de France / Università di Padova / University of Pavia 1566, 1571

Henricus BrucaeusCollège de France / Università di Bologna 1554, 1560

Christoph Glaser

Joseph BarthUniversität Wien 1772

Roger CotesUniversity of Cambridge 1706

Johannes CaseliusMartin-Luther-Universität Halle-Wittenberg / Universität Leipzig / Università di Pisa 1560, 1566

Cornelius MartiniUniversität Helmstedt 1592

Jacobus MartiniUniversität Helmstedt 1596

Ambrosius RhodiusMartin-Luther-Universität Halle-Wittenberg 1600, 1610

Paul RöberMartin-Luther-Universität Halle-Wittenberg 1613Martin-Luther-Universität Halle-Wittenberg 1617

Georg GroßhainMartin-Luther-Universität Halle-Wittenberg 1629

Franciscus (François Gomaer) GomarusCollège Saint-Guillaume à Strasbourg 1580Ruprecht-Karls-Universität Heidelberg 1585

Duncan LiddelUniversität Viadrina Frankfurt an der Oder / Universität Breslau / Universität Helmstedt 1582,1596

Jacobus (Jacob Harmensz.) ArminiusPhilipps-Universität Marburg / Universiteit Leiden 1582

Hieronymus (Girolamo Fabrici d'Acquapendente) FabriciusUniversità di Padova 1559

Salomon AlbertiMartin-Luther-Universität Halle-Wittenberg / Università di Padova 1564, 1574

Jan JesseniusUniversität Leipzig / Università di Padova 1588, 1591

Willebrord (Snel van Royen) SnelliusUniversiteit Leiden 1607

Thomas ErpeniusUniversiteit Leiden 1608

Moritz Valentin SteinmetzUniversität Leipzig 1550, 1567

Samuel Stryk

Petrus HoffveniusUniversiteit Leiden 1660

Jacob AndreaeEberhard-Karls-Universität Tübingen 1553

Giovanni Battista della MonteUniversità di Padova

Università degli Studi di Ferrara

Andreas (Andries van Wesel) VesaliusUniversité Catholique de Louvain / Università di Padova 1537

Pietro PomponazziUniversità di Padova 1487

Johannes (Johann Sturm) SturmiusUniversité Catholique de Louvain 1527

Jacques ToussainUniversité de Paris 1521

Adrien TurnèbeCollège de France 1532

Anton von StörckUniversität Wien 1757

Isaac NewtonUniversity of Cambridge 1668

Philipp MelanchthonRuprecht-Karls-Universität Heidelberg / Eberhard-Karls-Universität Tübingen 1511, 1514

Johannes HommelMartin-Luther-Universität Halle-Wittenberg 1543

Melchior JöstelMartin-Luther-Universität Halle-Wittenberg 1583, 1600

Ernestus HettenbachMartin-Luther-Universität Halle-Wittenberg 1576, 1591

Daniel SennertMartin-Luther-Universität Halle-Wittenberg 1594, 1599

Henricus ReneriusUniversiteit Leiden 1617

François Du Jon, Sr.

Johannes Polyander van KerckhovenRuprecht-Karls-Universität Heidelberg 1589

Université de Genève 1590

Paul WittichUniversität Leipzig / Martin-Luther-Universität Halle-Wittenberg 1566

Rudolph (Snel van Royen) SnelliusUniversität zu Köln / Ruprecht-Karls-Universität Heidelberg 1572

Gabriele FalloppioUniversità di Padova / Università degli Studi di Ferrara 1547

Ludolph van Ceulen

Joseph Justus ScaligerCollège de France 1563

Georg Joachim von Leuchen RheticusMartin-Luther-Universität Halle-Wittenberg 1535

Valentin (Valentinus Otho) OttoMartin-Luther-Universität Halle-Wittenberg 1570

Sebastian (Theodoricus) DietrichMartin-Luther-Universität Halle-Wittenberg 1544

Caspar PeucerMartin-Luther-Universität Halle-Wittenberg 1545

Johann HoffmannUniversity of Prague 1400Universität Leipzig 1414

Johannes Antonides van der LindenUniversiteit Franeker 1630

Jakob BeuerlinEberhard-Karls-Universität Tübingen 1551

Marco MusuroUniversità di Firenze 1486

Niccolò LeonicenoScuola Pubblica di Vicenza 1446

Università di Padova 1453

Antonio Musa BrasavolaUniversità degli Studi di Ferrara 1520

Nicoletto VerniaUniversità di Padova

Pietro RoccabonellaUniversità di Padova

Nicolas (Nicolaes Cleynaerts) ClénardUniversité Catholique de Louvain 1515, 1521

Johannes Winter von AndernachUniversité Catholique de Louvain / Collège de Tréguier 1527, 1532

Guillaume BudéUniversité d'Orléans / Université de Paris 1486, 1491

Melchior WolmarUniversité de Paris 1528

Gerard van SwietenUniversiteit Leiden 1725

Isaac BarrowUniversity of Cambridge 1652

Benjamin Pulleyn

Johannes StöfflerUniversität Ingolstadt 1476

Johann (Johannes Kapnion) ReuchlinUniversität Basel / Université de Poitiers 1477, 1481

Jan (Johannes Campensis) van CampenUniversité Catholique de Louvain / Universität Ingolstadt 1519

Andreas SchatoMartin-Luther-Universität Halle-Wittenberg 1562, 1578

Wolfgang FranzMartin-Luther-Universität Halle-Wittenberg 1590

Arnold SenguerdiusUniversiteit Leiden 1630

Johannes CalvinUniversité d'Orléans / Université de Bourges 1529, 1531

Lambert DaneauCollège de France 1553

Université d'Orléans / Université de Bourges 1559Université de Genève 1561

Valentin ThauUniversität Leipzig 1555

Valentine NaibodMartin-Luther-Universität Halle-Wittenberg / Universität Erfurt

Immanuel TremelliusUniversity of Cambridge 1549

Ruprecht-Karls-Universität Heidelberg 1561

Matteo Realdo (Renaldus Columbus) ColomboUniversità di Padova 1544

Johannes VolmarMartin-Luther-Universität Halle-Wittenberg 1515

Nicolaus (Mikołaj Kopernik) CopernicusUniwersytet Jagielloński / Università di Bologna / Università degli Studi di Ferrara / Università di Padova 1499

Menelaus WinsemiusUniversiteit Leiden 1613

Balthasar KaeuffelinEberhard-Karls-Universität Tübingen 1521

Janus LascarisUniversità di Padova 1472

Ognibene (Omnibonus Leonicenus) Bonisoli da LonigoUniversità di Mantova

Pelope

Gaetano da Thiene

Sigismondo Polcastro

Jacobus (Jacques Masson) LatomusCollège de Montaigu 1502

Katholieke Universiteit Leuven 1519

Rutger ResciusUniversité de Paris 1513

Jacobus (Jacques Dubois) SylviusUniversité de Paris / Université de Montpellier 1530

Georgius Hermonymus

Jacques (Jacobus Faber) Lefèvre d'ÉtaplesUniversité de Paris / Accademia Romana 1480

Vincenzo VivianiUniversità di Pisa 1642

Johannes ArgyropoulosUniversità di Padova 1444

Marsilio FicinoUniversità di Firenze 1462

Jacob ben Jehiel Loans

Heinrich Maius

Theodorus (Théodore de Bèze) BezaUniversité d'Orléans 1534, 1539

Antonius ThysiusUniversiteit Leiden 1582

Université de Genève / Ruprecht-Karls-Universität Heidelberg 1585, 1589

Andrea AlciatiUniversity of Pavia / Università di Bologna 1518

Erasmus ReinholdMartin-Luther-Universität Halle-Wittenberg 1535

Thomas CranmerUniversity of Cambridge 1515

Bonifazius ErasmiMartin-Luther-Universität Halle-Wittenberg 1509

Leonhard (Leonard Vitreatoris z Dobczyc) von DobschützUniwersytet Jagielloński 1489

Domenico Maria Novara da FerraraUniversità di Firenze 1483

Pieter (Petrus Pavius) PauwUniversiteit Leiden / Universität Rostock 1585

Basilios BessarionMystras 1436

Johannes Müller RegiomontanusUniversität Leipzig / Universität Wien 1457

Demetrios ChalcocondylesMystras / Accademia Romana 1452

Vittorino da FeltreUniversità di Padova 1416

Theodoros GazesConstantinople / Università di Mantova 1433

Jan StandonckCollège Sainte-Barbe / Collège de Montaigu 1474, 1490

Desiderius ErasmusCollège de Montaigu 1497 /1506

University of Turin 1506

Matthaeus Adrianus

Girolamo (Hieronymus Aleander) AleandroUniversità di Padova 1499, 1508

François DuboisUniversité de Paris 1516

Jean Tagault

Wolferd SenguerdiusAthenaeum Illustre Amsterdam / Universiteit Leiden 1666, 1667

Galileo GalileiUniversità di Pisa 1585

Benedetto CastelliUniversità di Padova 1610

Evangelista TorricelliUniversità di Roma La Sapienza

Jakob MilichAlbert-Ludwigs-Universität Freiburg im Breisgau / Universität Wien 1520, 1524

Gemma (Jemme Reinerszoon) FrisiusUniversité Catholique de Louvain 1529, 1536

Luca Pacioli

Georgios Plethon Gemistos 1380, 1393

Rudolf AgricolaUniversità degli Studi di Ferrara 1478

Guarino da Verona 1408

Moses Perez

Scipione FortiguerraUniversità di Firenze 1493

Ostilio RicciUniversita' di Brescia

Ulrich ZasiusAlbert-Ludwigs-Universität Freiburg im Breisgau 1501

Petrus (Pieter de Corte) CurtiusUniversité Catholique de Louvain 1513, 1530

Georg von PeuerbachUniversität Wien 1440

Demetrios Kydones

Manuel Chrysoloras

Elissaeus Judaeus

Angelo PolizianoUniversità di Firenze 1477

Nicolò Fontana Tartaglia

Alexander Hegius 1474

Maarten (Martinus Dorpius) van DorpUniversité Catholique de Louvain 1504, 1515

Johannes von GmundenUniversität Wien 1406

Nilos Kabasilas 1363

Cristoforo Landino

Thomas von Kempen à Kempis

Leo OutersUniversité Catholique de Louvain 1485

Heinrich von LangensteinUniversité de Paris 1363, 1375

Gregory Palamas

Anne du Bourg

Geert Gerardus Magnus Groote

Florens Florentius Radwyn Radewyns



Curvature-mediated Endocytosis

central process for interaction of cells with their environment

recent reports demonstrate that this process can happen ultrafast (ms)

at these time scales hydrodynamics are important, they may limit the speedand influence distribution of membrane molecules and cargo



But how to model a membrane with

curvature-inducing molecules in flow

?



1 Locally inextensible diffuse interface model

2 Including curvature-inducing membrane molecules

3 Numerical results

4 Summary



Biomembranes

Biomembranes in flow

interface is a lipid bilayer

lipids counteract bending ⇒ bending stiffness force δEδΓ

E[Γ] =

∫Γ

b

2(H −H0)2ds (Helfrich energy)

lipids resist any compression/extension ⇒ inextensibility:

∇Γ · v = 0 on Γ



global area constraint

Define Ω = Ω1 ∪ Γ ∪ Ω2.

inextensibility can be enforced globally instead of locally.

the globally constrained model reads

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[Γ]

δΓδΓ + λglobalHnδΓ in Ω

∇ · u =0 in Ω

∂tΓ =u|Γ

where the global Lagrange multiplier λglobal is obtained by requiring

d

dt

∫ΓdΓ =

∫ΓHu · n dΓ=0

leads to conserved membrane area, but can not prevent local membranestretching/compressionδEδΓ

= −b(∆ΓH + 1

2H(H2 −H2

0

))n

Inertia can be important in large/constricted blood vessels

4/25/14 10:48 AMPHOTO: Clogged artery - ABC News

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local inextensibility constraint

With local inextensibility constraints, the model reads:

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[Γ]

δΓδΓ +∇ · (δΓPλ) in Ω (1)

∇ · u =0 in Ω (2)

∂tΓ =u|Γ

where the Lagrange multiplier force takes the form of a surface tension

∇ · (δΓPλ) = δΓ∇Γλ− δΓnλH

and is obtained by requiring

P : ∇u = ∇Γ · u =0 (3)

with the surface projection

P =I− n⊗ n; n =∇φ|∇φ|

But how to discretize the inextensible Navier-Stokes system (1),(2),(3) ?



local inextensibility constraint

With local inextensibility constraints, the model reads:

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[Γ]

δΓδΓ +∇ · (δΓPλ) in Ω (1)

∇ · u =0 in Ω (2)

∂tΓ =u|Γ

where the Lagrange multiplier force takes the form of a surface tension

∇ · (δΓPλ) = δΓ∇Γλ− δΓnλH

and is obtained by requiring

P : ∇u = ∇Γ · u =0 (3)

with the surface projection

P =I− n⊗ n; n =∇φ|∇φ|

But how to discretize the inextensible Navier-Stokes system (1),(2),(3) ?



Diffuse interface model

Introduce a phase field φ torepresent the fluid domains

fixed interface thickness ε

Diffuse interface approximation of the bending energy [Du et al. Nonlin. 2005]:

E[φ] =

∫Ω

b

2ε

(ε∆φ−

1

ε(φ2 − 1)(φ+H0)

)2

dΩ.

Interface given byΓ = x|φ(x) = 0

Matched asymptotic expansion shows formal convergence

E[φ]→ E[Γ] for ε→ 0

.




The diffuse interface Navier-Stokes equation with global constraints (model A)[Du& Wang, JCP, 2006]

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ−λglobalf∇φ− λvolume∇φ in Ω

∇ · u =0 in Ω

Advected diffuse Willmore-flow equation to advect the interface

∂tφ+ u · ∇φ = −η(δE

δφ− λglobalf − λvolume

)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

where the global Lagrange multipliers λglobal, λvolume are obtained by requiring

d

dtV(φ) =

d

dt

∫Ω

1

2(φ+ 1) dΩ = 0 (volume constraint)

d

dtA(φ) =

d

dt

∫Ω

ε

2|∇φ|2 +

1

4ε(φ2 − 1)2 dΩ = 0. (global area constraint)

membrane surface area is conserved, but local membranestretching/compression still allowed




The diffuse interface Navier-Stokes equation with global constraints (model A)[Du& Wang, JCP, 2006]

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ−λglobalf∇φ− λvolume∇φ in Ω

∇ · u =0 in Ω


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

where the global Lagrange multipliers λglobal, λvolume are obtained by requiring

d

dtV(φ) =

d

dt

∫Ω

1

2(φ+ 1) dΩ = 0 (volume constraint)

d

dtA(φ) =

d

dt

∫Ω

ε

2|∇φ|2 +

1

4ε(φ2 − 1)2 dΩ = 0. (global area constraint)

membrane surface area is conserved, but local membranestretching/compression still allowed





3 Numerical results

4 Summary




Adding a local constraint, the diffuse interface model becomes (model B)[ Aland et al. (2014), J. Comput. Phys., in review. arXiv:1311.6558 ]

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ−λglobalf∇φ− λvolume∇φ

+∇ · (|∇φ|Pλ) in Ω

∇ · u =0 in Ω

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =0 in Ω (4)


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

matched asymptotic analysis shows convergence of (4) to inextensibilityconstraint P : ∇u = 0 on Γ and ∇ · (|∇φ|Pλ)→ ∇ · (δΓPλ) for ε→ 0




Adding a local constraint, the diffuse interface model becomes (model B)[ Aland et al. (2014), J. Comput. Phys., in review. arXiv:1311.6558 ]

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ−λglobalf∇φ− λvolume∇φ

+∇ · (|∇φ|Pλ) in Ω

∇ · u =0 in Ω

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =0 in Ω (4)


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

matched asymptotic analysis shows convergence of (4) to inextensibilityconstraint P : ∇u = 0 on Γ and ∇ · (|∇φ|Pλ)→ ∇ · (δΓPλ) for ε→ 0



Correction of cumulated errors: Relaxation

Over time small errors in inextensibility may accumulate and lead to localmembrane stretchingThe following mechanism corrects accumulated errors and drives the system backto the relaxed state

introduce a variable c to measure local stretching of the interface, initiallyc(x,0) = 1, evolved by

∂tc+ v · ∇c+ c∇Γ · v = 0 on Γ, (5)

locations where c deviates from 1 represent regions of compression (c > 1)and stretching (c < 1)

Hookes law gives strength of relaxation proportional to amount of localstretching:

P : ∇u = ζ(c− 1)/c on Γ

or in diffuse domain form:

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u = ζc− 1

c|∇φ| in Ω (6)

⇒ Model C



Correction of cumulated errors: Relaxation

To stabilize the interfacial advection equation

∂tc+ v · ∇c+ c∇Γ · v = 0 on Γ,

we add some small diffusion along the interface

∂tc+ v · ∇c+ c∇Γ · v − dT∆Γc = 0 on Γ. (7)

To discretize the equation it is extended off Γ using the diffuse domain approach[Ratz&Voigt, Comm.Math.Sci., (2006)]

∂t(|∇φ|c) +∇ · (|∇φ|vc)− dT∇(|∇φ|∇c) = 0 on Γ.

Additionally diffusion in normal direction can be added to extend c constant innormal direction off Γ

∂t(|∇φ|c) +∇ · (|∇φ|vc)− dT∇(|∇φ|∇c)− dN∇(|∇φ|nn · ∇c) = 0 on Γ.

matched asymptotic expansion shows convergence to (7) for ε→ 0

conservation of c along the interface is ensured:∫Ω |∇φ|c = const., even on

the discrete level



Governing Equations

Model A: standard model, only global inextensibility

Navier-Stokes equation to obtain the velocity

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ

− λglobalf∇φ− λvolume∇φ+∇ · (|∇φ|Pλ)

∇ · u =0

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =ζc− 1

c|∇φ|


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

An advection equation for the stretching measure c on the interface

∂t(|∇φ|c) +∇ · (|∇φ|uc)− dT∇ · (|∇φ|∇c)− dN∇ · (|∇φ|n⊗ n · ∇c) = 0



Governing Equations

Model B: with instantaneous local inextensibility


ρ(∂tu + u · ∇u)− ν∆u +∇p =δE[φ]

δφ∇φ


∇ · u =0

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =0ζc− 1

c|∇φ|


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)





Governing Equations

Model C: with instantaneous local inextensibility and relaxation


ρ(∂tu + u · ∇u)− ν∆u +∇p =dE[φ]

dφ∇φ


∇ · u =0

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =ζc− 1

c|∇φ|


∂tφ+ u · ∇φ = −η(δE


)δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)





Numerical treatment

C++ FEM library AMDiS

adaptive mesh

2D implementation

P2 elements except for P1 forpressure

direct solver: UMFPACK

semi-implicit time discretization

linearization by first orderTaylor expansion

remove stiffness due to bendingforces using monolithic, implicitsolvers (energy-stable-like)



Test scenario

We use a vesicle in shear flow to validate the model and investigate the influenceof local inextensibility.

Experimental results: Flow strength determines behavior: Tank treading(steady shape); Tumbling (non-steady rotating); Trembling (non-steady,non-rotating)



Test scenario

Numerical simulation of a vesicle in shear flow

Re=1/200

Be=20

H0 = 0

viscosity ratio: 10



Convergence I

Ev =

∫Ωε−1(1− φ2)2|∇Γ · v| ≈

∫Γ|∇Γ · v| (instantaneous stretching),

Figure: Convergence study showing super-linear decrease of theinstantaneous stretching Ev as a function of the interface thickness ε formodel B.



Convergence II

Ec =

∫Ωε−1(1− φ2)2|(c− 1)/c| ≈

∫Γ|(c− 1)/c| (accumulated stretching).

0.0150.030.0610

−5

10−4

10−3

10−2

ε

Ec

model Bquadraticlinear

0.0150.030.0610

−6

10−5

10−4

10−3

ε

Ec

model Cquadraticlinear

Figure: Convergence study showing super-linear decrease of Ec fordecreasing ε, for models B (left) and C (right).



Model Comparison I

Model comparison of the time evolution of a vesicle in shear flowmodel A model B model C

different time evolutions, although the vesicle shapes look identical

inextensibility in models B and C delays tumbling significantly



Model Comparison II

model A model B model C

Figure: The value of c along the vesicle interfaces for the different modelsat time t = 1.

.

model A shows strong compression at the tips, stretching at the sides

model B suppresses local stretching/compression significantly

model C shows no local stretching/compression



Model Comparison III

Ec =

∫Ωε−1(1− φ2)2|(c− 1)/c| ≈

∫Γ|(c− 1)/c| (accumulated stretching).

0 2 4 6 80

0.5

1

1.5

2

2.5

3

time

Ec

model Amodel Bmodel C

Figure: Comparison of accumulated stretching Ec for Models A (red), B(green) and C (blue)



Effect of inertia

0 2 4 6 8−2

−1

0

1

2

time

angl

e


0 2 4 6 80

0.5

1

1.5

time

angl

e


Figure: Model comparison of the inclination angle for Re = 1/200 (left).and Re = 1 (right).

Inertia suppresses tumbling.

(see also Salac & Miksis (2012); Laadhari et al. (2012); Kim & Lai (2012),..).



Interactions

Consider 2 vesicles in extensional flowmodel A

model B / model C





3 Numerical results

4 Summary



Bending energy including membrane molecules and flow

Let Γs ⊂ Γ be the molecule-covered part of the membrane.

The total bending energy is

E =

∫Γs

1

2bs (H −H0)2 dA︸︷︷︸

molecule bending

+

∫Γ

1

2bmH

2 dA︸︷︷︸membrane bending

+

∫Ω1∪Ω2

ρ

2|u|2dx︸︷︷︸

kinetic energy

, (8)

where bs and bm are bending moduli of molecule and clean membrane,respectively.

Introduce species concentration s(x, t) to keep track of Γs, where s = 1denotes complete coverage. Hence,

E =

∫Γ

1

2b(s) (H −H0(s))2 + Polynomial(s) dA (9)

Inextensibility and pure advection of s allow to drop the last term

Different choices for b(s) and H(s) possible

b(s) = (bs − bm) · s+ bm H0(s) = H0 · sb(s) =

(bs+bm)bm(bs+bm)−sbs

H0(s) = H0bs

bs+bms



Energy variation

Vary the energy by taking the time derivative

dtE =

∫Γb(s) (H −H0(s)) (dtH + n · ∇H u · n) +

b(s)

2(H −H0(s))2Hu · n

+∂E

∂sdts dA+

∫Ω1∪Ω2

ρdtu · u dx

where

∂E

∂s=

1

2b′(s)(H −H0(s))2 − b(s)(H −H0(s))H′0(s)

Assume the following balance laws for momentum and mass conservation:

ρ(∂tu + u · ∇u)−∇ · (νD) +∇p =0 in Ω1 ∪ Ω2 (Navier-Stokes)

∇ · u =0 in Ω1 ∪ Ω2 (incompressibility)

∇Γ · u =0 on Γ (inextensibility)

∂ts+ u · ∇s =0 on Γ (species advection)

where D = ∇u +∇uT .

Additionally assume the jump conditions for velocity and flow stress tensor

[−pI + νD]21 · n =F +∇Γλ−Hnλ

where F is yet unspecified interface force



Energy variation

plugging in balance laws and jump conditions into the energy timederivative, eventually leads to

dtE =

∫Γ−∆Γ (b(s)(H −H0(s)))u · n− b(s)(H −H0(s))‖∇Γn‖2u · n

+b(s)

2(H −H0(s))2 Hu · n−

∂E

∂s∇s · u + u · F dA

−∫

Ω1∪Ω2

ν

2|D|2 dx

2nd law of thermodynamics requires the surface intergral to vanish, hencethe choice

F =∆Γ (b(s)(H −H0(s)))n + b(s)(H −H0(s))‖∇Γn‖2n

−b(s)

2(H −H0(s))2Hn +

∂E

∂s∇s

gives decreasing energy

−∫

Ω1∪Ω2

ν

2|D|2 dx



sharp interface model

We obtain the sharp interface model

ρ(∂tu + u · ∇u)−∇ · (νD) +∇p =0 in Ω1 ∪ Ω2 (Navier-Stokes)

∇ · u =0 in Ω1 ∪ Ω2 (incompressibility)

∇Γ · u =0 on Γ (inextensibility)

∂ts+ u · ∇s =0 on Γ (species advection)

∂tΓ =n · u|Γ (interface advection)

[−pI + νD] · n = ∇Γλ−Hnλ+ ∆Γ (b(s)(H −H0(s)))n (jump condition)

+b(s)(H −H0(s))‖∇Γn‖2n

−b(s)

2(H −H0(s))2Hn +

∂E

∂s∇s



Diffuse energy variation

start with the energy

E =

∫Ω

1

2εb(s)

(ε∆φ−

1

ε(φ2 − 1)(φ+

√2εH0(s))

)2

+1

2ρ|u|2 dx

and assume similar balance laws

ρ(∂tu + u · ∇u)−∇ · (νD) +∇p =∇ · (|∇φ|Pλ) + F (10)

∇ · u =0 (11)

ξε2∇ · (φ2∇λ) + |∇φ|P : ∇u =0 (12)

∂ts+ u · ∇s =0 in Ω (13)

∂tφ+ u · ∇φ =− γg in Ω (14)

where F is yet unspecified interface force, g is an unspecified source

plug in balance laws into the energy time derivative to obtain F and g



Diffuse interface model for molecule-covered membrane

A locally inextensible Navier-Stokes equation to obtain the velocity

ρ(∂tu + u · ∇u)− ν∆u +∇p =δE

δφ∇φ+∇ · (|∇φ|Pλ)+

δE

δs∇s

∇ · u =0

ε2∇ · (φ2∇λ) + |∇φ|P : ∇u =ζc− 1

c|∇φ|

A diffuse Willmore-flow equation to advect the interface

∂tφ+ u · ∇φ = −ηδE

δφ

δE

δφ=

(∆(bf)−

1

ε2(3φ2 + 2H0φ− 1)bf

)f = ε∆φ−

1

ε(φ2 − 1)(φ+H0)

Instead of pure advection, we use an interfacial Cahn-Hilliard equation for themolecule concentration

∂t(|∇φ|s) +∇ · (|∇φ|us)− dT∇ · (|∇φ|∇µ)− dN∇ · (|∇φ|n⊗ n · ∇µ) =0

|∇φ|µ+ ε∇ · (|∇φ|∇s)− ε−1|∇φ|(4s3 − 6s2 + 2s) =0

An interfacial advection equation for the stretching measure c




Numerical treatment

C++ FEM library AMDiS

adaptive mesh

axisymmetric implementation

P2 elements except for P1 forpressure

direct solver: UMFPACK

semi-implicit time discretization

linearization by first orderTaylor expansion

non-stiff, implicitimplementation





3 Numerical results

4 Summary



Clathrin-mediated endocytosis

Clathrin

prototype of a curvature-inducingmolecule

bc = 12bm

H0 = (14nm)−1

Test setup

initially flat membrane with circularclathrin-covered region

axisymmetric equations

free stress BC for velocity

membrane pinned at the outerboundary at 90 angle



Clathrin-mediated endocytosis: first results

radius of clathrin region: 20nmspontaneous curvature of clathrin induces buddingno neck



Clathrin-mediated endocytosis: first results

radius of clathrin region: 20nmspontaneous curvature of clathrin induces buddingno neck



Clathrin-mediated endocytosis: larger clathrin region

radius of clathrin region: 30nmspontaneous curvature of clathrin induces budding and forms a neck



Clathrin-mediated endocytosis: larger clathrin region

radius of clathrin region: 30nmspontaneous curvature of clathrin induces budding and forms a neck



The effect of flow

With flow: small Willmore mobility such that the membrane is purely advected

fluid must be squeezed out of the vesicle to build a necktightening of the neck is inhibited by inextensibility

Without flow: large Willmore mobility is only driving force

membrane moves as if no fluids are presentneck tightens and vesicle finally pinches off flow results are in agreement with

biological observations, that an additional molecule (dynamin) is needed to triggerpinch-off



The effect of flow

With flow: small Willmore mobility such that the membrane is purely advected

fluid must be squeezed out of the vesicle to build a necktightening of the neck is inhibited by inextensibility

Without flow: large Willmore mobility is only driving force

membrane moves as if no fluids are presentneck tightens and vesicle finally pinches off flow results are in agreement with

biological observations, that an additional molecule (dynamin) is needed to triggerpinch-off



Varying the clathrin region size: stationary state

stationary shapes for various clathrin radii20nm 30nm 40nm 50nm 60nm



Varying the clathrin region size: dynamics

there is a critical (fastest) time for budding





3 Numerical results

4 Summary



Inextensibility model

Theoretical results

a local Lagrange multiplier was introduced to satisfy membraneinextensibility in diffuse interface methods (model B)

convergence to sharp inextensibility condition was shown numerically forε→ 0

asymptotic analysis shows convergence to ∇Γ · u = 0 for ε→ 0

a relaxation mechanism was proposed to drive the membrane back locally toan unstretched state (model C)

Numerical results

the effectiveness of the approach was shown in shear flow scenario

it was found that the treatment of the inextensibility constraint can cruciallyinfluence the vesicle dynamics

local inextensibility inhibits multiple vesicles from close contact

Aland et al.; Diffuse interface models for locally inextensible vesicles;arXiv:1311.6558



Inextensibility model: outlook

Outlook

include thermal fluctuations

apply the model to other nm-scale cell contexts, e.g. receptor-ligand bindingat cell-cell interfaces



Endocytosis model

Theoretical results

a diffuse and sharp interface model were derived for concentration-dependentbending stiffness and spont. curvature

derivation is in agreement with the 2nd law of thermodynamics

a modified diffuse interface equation is used to advect the moleculeconcentration on the membrane

an axisymmetric implementation was used to show the effectiveness of theapproach

Numerical results:

hydrodynamics allow a very fast building of a neck (in a few µs)

flow inhibits the tightening of the neck at a certain point

there is a critical (smallest) neck radius and a critical (fastest) budding time

Outlook

include attachment of bulk molecules to the membrane

explore the dependency of the arrival rate of membrane molecules on flow

explore intermediate dynamic shapes of the membrane and its influence onsoluble molecules and cargo

Thank you!



Endocytosis model

Theoretical results

a diffuse and sharp interface model were derived for concentration-dependentbending stiffness and spont. curvature

derivation is in agreement with the 2nd law of thermodynamics

a modified diffuse interface equation is used to advect the moleculeconcentration on the membrane

an axisymmetric implementation was used to show the effectiveness of theapproach

Numerical results:

hydrodynamics allow a very fast building of a neck (in a few µs)

flow inhibits the tightening of the neck at a certain point

there is a critical (smallest) neck radius and a critical (fastest) budding time

Outlook

include attachment of bulk molecules to the membrane

explore the dependency of the arrival rate of membrane molecules on flow

explore intermediate dynamic shapes of the membrane and its influence onsoluble molecules and cargo

Thank you!



Documents

Numerical simulation of endocytosis - UCLAhelper.ipam.ucla.edu/publications/mms2014/mms2014_11862.pdf · Nicholas Michael Katz Princeton University 1966 Dirk Jan Struik Universiteit