Table Of ContentLectu re Notes in
Biomathematics
Managing Editor: S. Levin
55
Modelling of Patterns
in Space and Time
Proceedings of a Workshop held by the
Sonderforschungsbereich 123 at Heidelberg
July 4-8, 1983
Edited by W Jager and J. D. Murray
Springer-Verlag
Berlin Heidelberg New York Tokyo 1984
Editorial Board
J. D. Cowan W Hirsch S. Karlin J. B. Keller M. Kimura
S. Levin (Managing Editor) R. C. Lewontin R. May J. D. Murray G. F. Oster
A. S. Perelson T. Poggio L. A. Segel
Editors
Willi Jager
Institut fUr Angewandte Mathematik
1m Neuenheimer Feld 294, 6900 Heidelberg, Federal Republic of Germany
James D. Murray
Mathematics Institute, Centre for Mathematical Biology
University of Oxford
24-29 St. Giles', Oxford OX1 3LB, Great Britain
AMS-MOS Subject Classification (1980): 34Cxx, 34Dxx, 35xx, 60Txx,
73Pxx, 76Exx, 76Zxx, 92A05, 92A09, 92A10, 92A15, 92A17, 92A40
ISBN-13: 978-3-540-13892-1 e-ISBN-13: 978-3-642-45589-6
001: 10.1007/978-3-642-45589-6
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks. Under
§ 54 of the German Copyright Law where copies are made for other than private use, a fee is
payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984
PREFACE
This volume contains a selection of papers presented at the work
shop "Modelling of Patterns in Space and Time", organized by the
80nderforschungsbereich 123, "8tochastische Mathematische Modelle", in
Heidelberg, July 4-8, 1983. The main aim of this workshop was to bring
together physicists, chemists, biologists and mathematicians for an
exchange of ideas and results in modelling patterns. Since the mathe
matical problems arising depend only partially on the particular field
of applications the interdisciplinary cooperation proved very useful.
The workshop mainly treated phenomena showing spatial structures. The
special areas covered were morphogenesis, growth in cell cultures,
competition systems, structured populations, chemotaxis, chemical
precipitation, space-time oscillations in chemical reactors, patterns
in flames and fluids and mathematical methods.
The discussions between experimentalists and theoreticians were
especially interesting and effective. The editors hope that these
proceedings reflect at least partially the atmosphere of this workshop.
For the convenience of the reader, the papers are ordered alpha
betically according to authors. However, the table of contents can
easily be grouped into the main topics of the workshop.
For practical reasons it was not possible to reproduce in colour the
beautiful pictures of patterns shown at the workshop. Since a larger
number of half-tone pictures could be included in this volume, the loss
of information has, however, been kept to a minimum.
The workshop has already stimulated cooperation between its parti
cipants and this volume is intended to spread this effect.
We would like to thank all participants that contributed to the
success of the workshop and also the authors of these proceedings who
have not only summarized their results but also initiated new research.
The assistance of P. GroBe in the preparation of this volume is also
gratefully appreciated. Last but not least, we acknowledge the support
of the Deutsche Forschungsgemeinschaft in sponsoring the workshop and
thus making this volume possible.
Heidelberg, July 1984 The editors
LIST OF AUTHORS
ALT, W.; Sonderforschunqsbereich 123, Universitat Heidelberg, 1m Neuen
heimer Feld 293, D-6900 Heidelberg
AVNIR, D.; Department of Organic Chemistry, Hebrew University of
Jerusalem, Jerusalem 91904, Israel
BOON, J.-P.; Faculte des Sciences, C.P. 231, Universite Libre de
Bruxelles, B-1050 Bruxelles, Belgium
BROWN, R.A.; Department of Chemical Engineering and Materials Processing
Center, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA
BURGESS, A.E.; Department of Chemistry, Glasgow College of Technology,
Cowcaddens Road, Glasgow G4 OBA, U.K.
BUSSE, F.H.; Department of Earth and Space Sciences and Institute of
Geophysics and Planetary Physics, University of California, Los
Angeles, CA 90024, USA
CHILDRESS, St.; Courant Institute of Mathematical Sciences, New York
University, New York, N.Y. 10012, USA
COOMBS, J.P.; Department of Microbiology, University College, Newport
Road, Cardiff DF2 1TA, Wales
ERNEUX, Th.; Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, Illinois 60201, USA
FIFE, P.C.; Department of Mathematics, University of Arizona, Tucson,
Arizona 85721, USA
GOMATAM, J.; Department of Mathematics, Glasgow College of Technology,
Cowcaddens Road, Glasgow G4 OBA, U.K.
GYLLENBERG, M.; Helsinki University of Technology, Institute of
Mechanics, SF-02150 Espoo 15, Finland
HARRIS, A.K.; Department of Biology, Wilson Hall (046A), University of
North Carolina, Chapel Hill, North Carolina 27514, USA
HERPIGNY, B.; Faculte des Sciences, C.P. 231, Universite Libre de
Bruxelles, B-1050 Bruxelles, Belgium
HOPPENSTEADT, F.C.; Department of Mathematics, University of Utah, Salt
Lake City, Utah 84112, USA
HUDSON, J.L.; Department of Chemical Engineering, University of Virginia,
Charlottesville, VA 22901, USA
JAFFE, S.; Department of Microbiology, University College, Newport Road,
Cardiff CF2 1TA, Wales
J~GER, W.; Institut fUr Angewandte Mathematik, Universitat Heidelberg,
1m Neuenheimer Feld 294, D-6900 Heidelberg
KAGAN, M.C.; Department of Organic Chemistry, Hebrew University of
Jerusalem, Jerusalem 91904, Israel
v
KEENER, J.P.; Department of Mathematics, University of Utah, Salt Lake
City, Utah 84112, USA
KEMMNER, W.; Institut fur Zoologie II, Universitat Heidelberg, Im
Neuenheimer Feld 230, D-6900 Heidelberg
KESHET, Y.; Department of Applied Mathematics, Weizmann Institute of
Science, 76100 Rehovot, Israel
LAUFFENBURGER, D.A.; Department of Chemical Engineering, University of
Pennsylvania, Philadelphia, PA 19104, USA
MAREK, M.; Prague Institute of Chemical Technology, 16628 Prague 6, C~S SR
MATKOWSKY, B.J.; Department of Engineering Sciences and Applied
Mathematics, Northwestern University, Evanston, Illinois 60201, USA
MEINHARDT, H.; Max-Planck-Institut fur Virusforschung, D-7400 Tubingen
MEISELS, E.; Department of Organic Chemistry, Hebrew University of
Jerusalem, Jerusalem 91904, Israel
MULLER, S.C.; Max-Planck-Institut fur Ernahrungsphysiologie, Rheinland
damm 201, D-4600 Dortmund
MURRAY, J.D.; Centre for Mathematical Biology, Mathematical Institute,
University of Oxford, Oxford OX1 3LB, England
NICOLAENKO, D.; Los Alamos National Lab., Los Alamos, USA
NISHIURA, Y.; Kyoto Sangyo University, Kyoto 603, Japan
ODELL, G.M.; Department of Mathematics, Rensellaer Polytechnic Institute,
Troy, N.Y. 12181, USA
OSTER, G.F.; Department of Biophysics, University of California,
Berkeley, CA 94720, USA
PELEG, S.; Department of Computer Science, Hebrew University of Jerusalem,
Jerusalem 91904, Israel
PLESSER, Th.; Max-Planck-Institut fur Ernahrungsphysiologie, Rheinland
damm 201, D-4600 Dortmund
POPPE, Ch.; Sonderforschungsbereich 123, Universitat Heidelberg, Im
Neuenheimer Feld 293, D-6900 Heidelberg
REISS, E.L.; Department of Engineering Sciences and Applied Mathematics,
Northwestern University, Evanston, Illinois 60201, USA
ROSSLER, O.E.; Institute for Physical and Theoretical Chemistry,
University of Tubingen, D-7400 Tubingen
ROSEN, R.; Department of Physiology and Biophysics, Dalhousie University,
Halifax, N.S., Canada B3H 4H7
ROTHE, F.; Lehrstuhl fur Biomathematik, Universitat Tubingen, Auf der
Morgenstelle 28, D-7400 Tubingen
SCHAAF, R.; Sonderforschungsbereich 123, Universitat Heidelberg, Im
Neuenheimer Feld 293, D-6900 Heidelberg
SEGEL, L.A.; Department of Applied Mathematics, Weizmann Institute of
Science, 76100 Rehovot, Israel
SHI, J.S.; Department of Mathematics, University of Michigan, Ann Arbor,
Michigan 48109, USA
SMaLLER, J.A.; Department of Mathematics, University of Michigan, Ann
Arbor, Michigan 48109, USA
TAUTU, P.; Department of Mathematical Models, Institute of Documentation,
Information and Statistics, German Cancer Research Center,
D-6900 Heidelberg
UNGAR, L.A.; Department of Chemical Engineering and Materials
Processing Center, Massachusetts Institute of Technology, Cambridge,
MA 02139, USA
VENZL, G.; Institut fur Theoretische Physik, Physik-Department der
Technischen Universitat Munchen, D-8046 Garching
WELSH, B.J.; Department of Mathematics, Glasgow College of Technology,
Cowcaddens Road, Glasgow G4 OBA, U.K.
WIMPENNY, J.; Department of Microbiology, University College, Newport
Road, Cardiff CF2 1TA, Wales
TABLE OF CONTENTS
ALT, W., Contraction patterns in a viscous polymer system ••..••.•••
AVNIR, D., s. KAGAN, M.L.
BOON, J.P., B. Herpigny, Formation and evolution of spatial struc
tures in chemotactic bacterial systems .•••..••.••..•••.•..•..•. 13
BROWN, R.A., L.A. Ungar, Pattern formation in directional solidifi
cation: the nonlinear evolution of cellular melt/solid inter-
faces. • . . . . • • . . • . . . • • • • • • • • • • . . • • • • • • . . . . • . • • • • . • . • • • . • . . • • • . .. 30
BURGESS, A.E., B.J. Welsh, J. Gornatam, Evolution of 3-D chemical
waves in the BZ reaction medium................................ 43
BUSSE, F.H., Patterns of bifurcations in plane layers and spherical
shells......................................................... 51
CHILDRESS, St., Chemotactic collapse in two dimensions ..•....••.••. 61
COOMBS, J.P., s. WIMPENNY, J.W.T.
ERNEUX, Th., B.J. Matkowsky, E.L. Reiss, Singular bifurcation in
reaction-diffusion systems .•.....••.•.•.••.••.••.•.••.••••••... 67
FIFE, P.C., B. Nicolaenko, How chemical structure determines spatial
structure in flame profiles •..•••.•..•.••.•.••••.•..•.•.•••.••• 73
GOMATAM, J., s. BURGESS, A.E.
GYLLENBERG, M., An age-dependent population model with applications
to microbial growth processes .•..•.•..••....••....•..•.•.••.••. 87
HARRIS, A.K., Cell traction and the generation of anatomical struc-
ture ..•.•..•....••.•..••..••.•.....••.••..•..•.......••..•..•.. 103
HERPIGNY, B., s. BOON, J.P.
HOPPENSTEADT, F.C., W. Jager, Ch. Poppe, A hysteresis model for
bacterial growth patterns...................................... 123
HUDSON, J.L., O.E. Rossler, Chaos in simple three- and four-variable
chemical systems............................................... 135
JAFFE, S., s. WIMPENNY, J.W.T.
J~GER, W., s. HOPPENSTEADT, F.C.
KAGAN, J.L., S. Peleg, E. Meisels, D. Avnir, Spatial structures
induced by chemical reactions at interfaces: survey of some
possible models and computerized pattern analysis ••••..•.•••... 146
KEENER, J.P., Dynamic patterns in excitable media •.•••..•.•..•....• 157
KEMMNER, W., Head regeneration in hydra: biological studies and a
model •••....••••••••••.•.••.•..••••.....•..••.•.•.•.••.••..•••• 170
KESHET, Y., L.A. Segel, Pattern formation in aspect •••.•••.••...••. 188
LAUFFENBURGER, D.A., Chemotaxis and cell aggregation ..•..•••••..•.• 198
MAREK, M., Turing structures, periodic and chaotic regimes in
coupled cells.................................................. 214
VIII
MATKOWSKY, B.J., s. ERNEUX, Th.
MEINHARDT, H., Digits, segments, somites - the superposition of
sequential and periodic structures •.•...•.••.......•..••......• 228
MEISELS, E., s. KAGAN, M.L.
MULLER, S.C., Th. PLESSER, Spatial pattern formation in thin layers
of NADH-solutions .••.••..•........•.•..•..•.••.....••.......... 246
MULLER, S.C., G. VENZL, Pattern formation in precipitation
processes .•...••••..•....••..•..•....•....•...•.•..•......•.... 254
MURRAY, J.D., On a mechanical model for morphogenesis: mesenchymal
patterns. . . .• .• • . . . . • • . . • . . . . . • . . . • . . . . . . . . . . . . . . . •. . . . . • . . • . .• 279
NICOLAENKO, D., s. FIFE, P.C.
NISHIURA, Y., Every multi-mode singularly perturbed solution
recovers its stability - from a global bifurcation view point •. 292
ODELL, G.M., s. OSTER, G.F.
OSTER, G.F., ODELL, G.M., A mechanochemical model for plasmodial
oscillations in physarum....................................... 302
PELEG, S., s. KAGAN, M.L.
PLESSER, s. MULLER, S.C.
P6PPE, Ch., s. HOPPENSTEADT, F.C.
REISS, E.L., s. ERNEUX, Th.
R6SSLER, O.E., s. HUDSON, J.L.
ROSEN, R., Genomic control of global features in morphogenesis ..... 318
ROTHE, F., Patterns of starvation in a distributed predator-prey
system •.......•...•.•...•...•.• ,. . .. .•...•..• ... ... .. ..• ... ..•. 331
SCHAAF, R., Global branches of one dimensional stationary solutions
to chemotaxis systems and stability •.............•......••..... 341
SEGEL, L.A., s. KESHET, Y.
SHI, J.S., s. SMOLLER, J.A.
SMOLLER, J.A., SHI, J.S., Analytical and topological methods for
reaction-diffusion equations................................... 350
TAUTU, P., Branching processes with interaction as models of
cellular pattern formation..................................... 364
UNGAR, L.A., s. BROWN, R.A.
VENZL, G., s. MULLER, S.C.
WELSH, B.J., s. BURGESS, A.E.
WIMPENNY, J.W.T., S. JAFFE, J.P. COOMBS, Periodic growth phenomena
in spatially organized microbial systems •....••..•....•.•...•.• 388
CONTRACTION PATTERNS IN A VISCOUS POLYMER SYSTEM
Wolfgang Alt
Sonderforschungsbereich 123
Universitat Heidelberg
1m Neuenheimer Feld 293
0-6900 Heidelberg
The explanation of cell locomotion is one of the current biological
research topics. During the last decade it became apparent that the ac
tive forces which deform the shape of a motile cell (amoeboid cell, fi
broblast or leukocyte, for example) and which finally lead to its de
placement are provided by contractile filaments, being present within
the ceil plasma, in particular near the cell membrane.
Amoeboid locomotion,see figurre] 1a, has been studied extensively. The
mathematical model of G. Odell describes the interaction and trans
port of filaments in an endoplasmatic flow governed by sol-gel-transitions
within various pseudopods of a moving cell. Further extensions of this
model include the viscoelastic properties of actomyosin gels and the
control by diffusing substances (like ca++), see the recent work of
G. Oster, G. Odell [9] in this volume.
Figure la.
Amoeboid locomotion,
cpo [1].
Figure lb.
LeUkocyte locomotion,
cpo [10]
2
The locomotion of fibroblasts or leukocytes, for example, seems to
be based on similar, but slightly different mechanisms, see figure lb.
Periodical protrusion of flat membrane extensions (lamellipods), their
attachment to some underlying substrate and their partial withdrawal
opposite to the direction of cell displacement (ruffling) suggest that
actin filaments, distributed inside the (hyalo-)plasma of the lammelli
podium, and their interactions play the most important role, see [8].
In order to understand the possible function of the contractile acto
myosin polymer system itself, simple mathematical models should be in
vestigated, which are able to reproduce the following observed phenomena:
(a) formation of (at least transiently) stable contraction centers,
(b) oscillatory competition between different contraction centers.
(4]
In a first attempt M. Dembo, F. Harlow and the author proposed
a model for a highly viscous "fluid" of actin bundles (or filaments)
which attract each other via binding to myosin polymers. The actin
bundles are formed (nucleated and polymerized) from a large reservoir
of actin monomers, but when density increases they are depolymerized
and disassembled. Assuming that mutual attraction, shearing forces (vis
cosity) and friction dominate other forces (inertia, or elasticity as
[9J)
in we get the following simplified balance equations for the ~
concentration u(t,x) and the mean velocity v(t,x) of polymer bundles,
for more details see [2]
Mass balance:
(1)
where the kinetic growth function
flu) = N (u) - u·N (u)
+ -
contains a density dependent nucleation term N+(U) and a disassembly
rate N_(U). Typically f has exactly one zero, without restriction at
u=l, and f decreases for u~l with
f I (1) = -n < 0 •
u
Figure 2. Kinetic growth
function f as in (19) with
p=3, n=4.
