The Mathematical Intelligencer Vol 35 No 1 March 2013

88 Pages·2013·8.64 MB·English

Letter
A Discrepancy RR
obert Jones’s article, ‘‘Felix Hausdorff in Bonn,’’ in
the Spring 2012 issue of The Mathematical Intelli-
gencer was engrossing, but puzzling, too.
P R According to the article, Hausdorff, his wife, and her
ETER OSS
sister committed suicide in 1942 rather than face concen-
tration camps. But the translation of the inscription on the
tombstone gives ‘‘29. Januar’’ for the death of the sister,
Edith, whereas Hausdorff and his wife both died on ‘‘26.
The Mathematical Intelligencer encourages comments Januar.’’ If this is not a typographical error, then some
explanation should be given for the 3-day delay of the
about the material in this issue. Letters to the editor
sister’s death.
I’m sure there are many other discerning readers of the
should be sent to the editor-in-chief.
Intelligencerwhowouldappreciateanexplanationforthis
little riddle.
DepartmentofMathematicsandComputerScience(retired)
SantaClaraUniversity
SantaClara,CA95053-0290
USA
e-mail:[email protected]
The Editor Replies
Thank you for pointing out this discrepancy; we agree it
should be clarified.
In Mathematicians Under the Nazis, Princeton Univer-
sity Press, 2003, p. 459, Sanford Segal explains: ‘‘That
evening they took lethal does of barbiturates (veronal); in
the morning Hausdorff and his wife were dead; her sister
lingered for a few days in a coma.’’ For more about the
Hausdorffs’ decision to end their lives, see also Tran-
scending Tradition: Jewish Mathematicians in German-
Speaking Academic Culture, reviewed by Reinhard
Siegmund-Schulze in this issue.
1
(cid:2)2012SpringerScience+BusinessMediaNewYork,Volume35,Number1,2013
DOI10.1007/s00283-012-9343-7
Letter
Three Thoughts provedbycontradiction.’’Notso,sayKarinUsadiKatzand
Mikhail Katz! They write [3, pages 13–14],
p
Without exploiting the hypothetical equality ffi2ffiffi¼m,
on ‘‘Prime n
onecanexhibitpositivelowerboundsforthedifference
p
j ffi2ffiffi(cid:2)mj in terms of the denominator n, resulting in a
n
Simplicity’’ constructively adequate proof of irrationality.
In a footnote, they give the lower bound 1/(3n2).
p
ffiffiffi
Fortunately, our statement about 2 was in no way
essential to our theses.
M H
ICHAEL ARDY
Chronology
The Mathematical Intelligencer encourages comments
Finally, I would like to clarify something that might be
aboutthematerialinthisissue.Lettersshouldbesentto confusing. The paper as submitted to this journal con-
trasted a passage [1, pages 122–123] written by G.
eitheroftheeditors-in-chief,ChandlerDavisorMarjorie H. Hardy more than a hundred years ago with ‘‘Euclid’s
proof as presented by Øystein Ore above.’’ The word
Senechal.
‘‘above’’ meant earlier in our paper, where Ore’s para-
phrase [5, page 65] of the proof was quoted in its entirety.
II Some copyeditor changed ‘‘above’’ to ‘‘earlier.’’ I objected
n 2009, Catherine Woodgold and I published ‘‘Prime
to the change on the grounds that it makes it appear that
Simplicity’’[2],examiningthebeliefthatEuclid’sfamous
we were saying Ore’s 1948 book appeared earlier than G.
proof of the infinitude of prime numbers was by
H. Hardy’s 1908 book. I was told that ‘‘above’’ would
contradiction. We demonstrated that that belief is wide-
appear in the published paper. It didn’t. For the record,
spreadamongmathematiciansandisfalse:Euclid’sproofis
my coauthor and I were aware that 1948 is not earlier
simpler and better than the frequently seen proof by
than 1908.
contradiction. The extra complication of the indirect proof
serves no purpose and has pitfalls that can mislead the
reader.
REFERENCES
[1] Hardy,G.H.,ACourseofPureMathematics,CambridgeUniversity
Dirichlet
Press,1908.
The many examples we cited were all from sources since
[2] Hardy, M. and Woodgold, C., ‘‘Prime Simplicity,’’ Mathematical
1900.Thiscutoffdatewasnotplanned.Wesetouttodocu-
Intelligencer31(2009),no.4,44–52.
mentmodernviews.Ifwehadsetouttotracethehistoryof
[3] Katz,K.U.andKatz,M.,‘‘MeaninginClassicalMathematics:IsItat
the misunderstanding, we might not have missed a gem
pointedoutbyRobertJ.Gray:likemanylaterauthors,J.P.G. OddswithIntuitionism?’’\http://arxiv.org/pdf/1110.5456v1.pdf[.
Lejeune-Dirichlet, in a posthumous book [4, pages 9–10], [4] Lejeune-Dirichlet,J.P.G.,LecturesonNumberTheory,American
falselyattributedtheproofbycontradictiontoEuclid.Could MathematicalSociety,1999(translationbyJohnStillwellofVorles-
all those twentieth-century occurrences of the error stem ungenu¨berZahlentheorie,FriedrichViewegundSohn,1863).
fromDirichlet?ThatquestionIleaveopen. [5] Ore, Ø., Number Theory and Its History, Courier Dover Publica-
tions, 1988 (reprint of a book published by McGraw–Hill in
1948).
Square Roots and Contradictions
WenotedthatneitherwenorEuclidobjectedtoproofsby DepartmentofMathematicsandStatistics
contradiction in general, and in particular Euclid proved St.CloudStateUniversity
p
ffiffiffi
the irrationality of 2 by contradiction. Later, on page 46, St.Cloud,MN56301,USA
we said that that fact is ‘‘a negative result that can only be e-mail:[email protected]
2
THEMATHEMATICALINTELLIGENCER(cid:2)2012SpringerScience+BusinessMedia,LLC
DOI10.1007/s00283-012-9322-z
Words about Numbers
Nicole Yunger Halpern
How do you
measure a line?
In feet or meters,
meter or rhyme?
Do rhythms
or do algorithms
better keep
your thoughts with them?
Could you draw
figures of speech
if you had
a compass in reach?
If armed with compass,
quotes, and quips,
would you make an ellipsis
or an ellipse?
Ellipses eclipsed
turn hyperbolic;
do hyperboles
turn ironic?
From irony
could you make steel?
Does wordsmithing have
more appeal?
Do you read proofs
or proofread reads?
Do quarter-rhymes
half-suit your needs?
Lovers of puns,
pi, and logs:
When did you last
dialogue?
PerimeterInstituteforTheoreticalPhysics
Waterloo,ONN2L2Y5
Canada
e-mail:[email protected]
3
(cid:2)2012SpringerScience+BusinessMedia,LLC,Volume35,Number1,2013
DOI10.1007/s00283-012-9335-7
The Colors of Math
Gizem Karaali
DepartmentofMathematics,
PomonaCollege
610NorthCollegeAvenue
Claremont,CA91711
USA
e-mail:[email protected]
4
THEMATHEMATICALINTELLIGENCER(cid:2)2012SpringerScience+BusinessMedia,LLC
DOI10.1007/s00283-012-9327-7
Death and Mathematics
Poems
Reuben Hersh
GEORGE PO´LYA DUTCH MATHEMATICIANS
was blind, his last few years. Van der Waerden thinks best keep his Chair in Leipzig
Kolmogorov too. till the War is over.
Lie down, lie down, old plowman. The Dutch did not forgive him.
Life is cruel, death is kind. Brouwer thinks best serve the Germans,
And deaf and dumb. disliking French even more than Jews.
And blind. The Dutch did not forgive him.
Freudenthal, Dutch Jew born German,
hidden Underground, writes a prize-winning novel.
SONIA AND EMMY
At 85 is found,
died too soon. seated on a park bench,
Karl burned Sonia’s letters. among playing children.
On Bryn Mawr grounds are buried
Emmy’s ashes.
ANANALYST,ALOGICIAN,ANDTHREETOPOLOGISTS
Herbrand and Galicki, climbing in the Alps.
OSWALD TEICHMU¨LLER
Paley, skiing in the Rockies.
Born 1913.
Urysohn, swimming off Brittany.
In 1933
Hurewicz, standing on Uxmal in Yucata´n.
his gang bars Landau from the platform.
From Stalingrad in 1943
his corpse does not return.
PHYLLIS’S ASHES
Ten years ago she planted here these tulips,
that now feed on her ashes.
BELLA ABRAMOVNA SUBBOTOVSKAYA
Where are the grasses now, that in ten years
barely remembered, mostly forgotten.
my ashes will be feeding?
just found classrooms, found teachers, provided chalk.
even made sandwiches.
a woman easily disposed of, easily executed.
1000CaminoRancheros
SantaFe,NM87505
USA
e-mail:[email protected]
5
(cid:2)2013SpringerScience+BusinessMedia,LLC,Volume35,Number1,2013
DOI10.1007/s00283-012-9347-3
Viewpoint
Is Mathematics JJointly with another mathematician, a biophysicist, and
two diabetes doctors, I released a textbook, BetaSys –
Systems Biology of Regulated Exocytosis in Pancreatic
Invading Human
b-Cells,1 inwhichabroadinternationalteamsummarizesthe
state of our current understanding of the cell-physiological
Cells? Impressions events accompanying both successful and impaired insulin
secretion.InthisViewpoint,Idescribesomeofmyexperiences
asamathematiciancooperatingwithdiabetesspecialists,and
from a Collaboration
thewiderquestionsthatthoseexperiencesraise.
Advanced Equipment and Basic Ignorance
with Diabetes Doctors
Along with space exploration and military and civilian
nuclearpowerdesign,medicaldevicesbelongtothemath-
ematically most sophisticated areas of modern technology.
B B -B
ERNHELM OOSS AVNBEK Many mathematicians have or could have contributed to
magneticspinresonanceimaging(MRI),andthereishardlya
singlemathematicianwhomasterstheentiremathinvolved
in that technology. The same goes for electron tomogra-
phy,multibeamconfocallasermicroscopy,andmanyother
TheViewpointcolumnoffersreadersofThe
advanceddevices.Medicinehasbecomeamathematicaldis-
MathematicalIntelligencertheopportunitytowriteabout cipline.Theominousmilitary–industrialcomplexhasmetasta-
sized; an eminently mathematical sickness-and-health
anyissueofinteresttotheinternationalmathematical industryhasgrownupalongsideit.
Butmathematicsisencapsulatedintheapparatus.Whe-
community.Disagreementandcontroversyarewelcome.
theritisaboutaspecificdiagnosisortreatment,mostpatients,
Theviewsandopinionsexpressedhere,however,are at least those who are mathematicalphysicists,willbe sur-
prisedathowlittlemedicalsciencereallyseemstoknowand
exclusivelythoseoftheauthor,andthepublisherand understandaboutparticulardiseases.Itisquitenormalthata
doctor must simply experiment – or just stick to an estab-
editors-in-chiefdonotendorsethemoraccept
lishedsymptomdiagnosisandsymptomtreatment.Withouta
detailed identification of the real causes of the individual
responsibilityforthem.Viewpointshouldbesubmittedto
patient’sailment,oftenasuccessfultreatment,definedasa
the editor-in-chief, Marjorie Senechal. cure,isunattainable.
Physicscanalsobecomplicatedandinmanycaseswithout
establishedanswers.Butinphysicsthereisafterallonlyavery
shortlistof‘‘FirstPrinciples’’thatonemuststickto.Therewe
have relatively well-defined interfaces between established
knowledge,reasonedorvaguepresumption,andignorance.
Andinmostcases,ourignoranceinphysicscanbecondensed
insomemathematicalequations(whichwemaynotimme-
diatelyfullyunderstand).Thisisnotsoinmedicine.
Challenges (for Mathematicians) in Cell
Research
The Strong Medical Pull
From pure mathematical research, we know the feeling of
being pulled forward by an overarching issue: the
1Booß-Bavnbek,B.;Klo¨sgen,B.;Larsen,J.;Pociot,F.;Renstro¨m,E.(eds.),BetaSys–SystemsBiologyofRegulatedExocytosisinPancreaticb-Cells,series:Systems
Biology,Springer,Berlin-Heidelberg-NewYork,2011,XVIII,558pages,104illustr.,53incolor.Withonlinevideosandupdates.ISBN978-1-4419-6955-2.
[1]ComprehensivereviewinDiabetologia,DOI10.1007/s00125-011-2269-3.Inthefollowingthismonographwillbecitedas[1]
6
THEMATHEMATICALINTELLIGENCER(cid:2)2013SpringerScience+BusinessMediaNewYork
DOI10.1007/s00283-012-9349-1
relationship between local and global properties, between
thesmoothandthecontinuous,betweenanalyticandalge-
braic methods, the Four-Color Problem, the Poincare´
Conjecture, the Riemann Hypothesis, the Clay Millennium
Problems.Ofcourse,wewouldneveradmitsuchpersonal
ambitionsinpublic.Buttomethereisnodoubtabouttherole
thatmajorwell-statedproblemsplayandhaveplayedinthe
designofthecareerpathsofmanymathematicians,atleast
indirectly and in daydreams: with many doubts and a per-
sistent feeling of self-deception and of fighting against
mountains–orwindmills.
Workingasamathematicianwithdiabetesdoctorsisdif-
ferent.Abristlingcascadeofmedicalissuespullstheresearch
forward: For nearly 90 years we have known that lack of
secretionofthehormoneinsulinisoneofthemanyserious
issuesinbothdiabetestype1(juvenile)andtype2(obesity
Figure 1. Cartoon of a pancreatic b-cell with glucose-trig-
andagedriven).Foralargegroupofthesepatients,insulinis
gered insulin secretion. After Renstro¨m (2011) in ref. [1] in
actuallyproducedintheirpancreaticb-cells(Fig. 1),anditis
footnote1,p.37,reproducedwithpermission.Originalfigure
storedinthousandsofminibags,vesicles,inthecell’sinterior
(cid:2)Springer-Verlag.
(Fig. 2). But the cells do not respond correctly to external
stimuliwiththeactualsecretion,calledregulatedexocytosis.
That manifests itself in elevated blood sugar, which for
4000 yearshasbeentestedandmeasuredbyurinesamples.2
Wecallitasymptomdiagnosis,becausethediagnosissays
nothingaboutthewiderangeofcausesthatmayunderliethe
lackofuptakeofglucoseinthemuscles.
Previously,failureofinsulinsecretionautomaticallyledto
weakeningofthemuscles,inflammationoftheextremities,
lossofvision,andthebody’sfinaldecay.Sincethediscovery
of insulin, this tragic developmentcan be countered by an
artificialsupplyofinsulinbyinjectionseveraltimesaday.We
callitasymptomtreatment,becauseitisnotevenanattempt
..........................................................................
R BERNHELMBOOSS-BAVNBEK lecturesat
O
H Roskilde University (Denmark) on spectral
T
U problems of global analysis and emerging
A
geometrical problemsin cell physiology.His
Figure2. Electron tomographic image of pancreatic b-cell.
recent books include Index Theory with
Details of cellular anatomy marked by false-colour coding.
Applications to Mathematics and Physics
Note insulin-containing granules entered in blue and nucleus
(International Press, 2013, inprint), New
in yellow(courtesy B.Marsh).
PathsTowardsQuantumGravity(Lecture
Notes in Physics 807, Springer, 2010) and
tocurethepatientortomakeanefforttorestorethebody’s
BetaSys: Systems Biology of Regulated
owninsulinsecretion.Someclaimthattherelativesuccessof
ExocytosisinPancreatic b-Cells (Springer,
theoverallsymptomdiagnosisandsymptomatictreatmentof
2011).
diabeteshasblockedpatient-centered,individualizeddiag-
nosisandtreatment.
Department of Science, Systems and
In any case, in collaborating with diabetes doctors, a
Modelling/IMFUFA–In, With and
mathematician is continually pulled forward by well-
About Mathematics and Physics defined medical problems. In this case, the problem is to
Roskilde University detectthefunctioningandsystembehavioroftheregulated
Universitetsvej 1, P.O. Box 260 exocytosisinhealthyb-cellsandtoidentifyeverythingthat
DK-4000 Roskilde can stand in the way in the case of weakened b-cells.
Denmark Doctors hope that mathematicians will help find a way to
e-mail: [email protected] an earlier and more specific diagnosis, or even a cure or
2Theearliestpreservedreport(inBendexEbbell’sCopenhageninterpretationof1937)isfromtheEgyptEbersPapyrusof1536BCE,instruction197,column39,line7,
reproducedinallitsambiguityonhttp://biology.bard.edu/ferguson/course/bio407/Carpenter_et_al_(1998).pdf.
7
(cid:2)2013SpringerScience+BusinessMediaNewYork,Volume35,Number1,2013
fiddledperceptionsofunconfinedcreativity,a` la:‘‘itshould
probably be the cell nucleus that controls the process,’’ or
‘‘there is a certain rate, which determines the transition
betweenonestageandthenext,’’or‘‘acorrelationbetween
the one process and another process is unquestionable.’’
Explanationsholduntilovertakenbynewdataandwillthen
be‘‘adjusted.’’Theywillneverbefalsified,becausetheyare
freestandingandvariableandnot,asinphysics,tiedtofirst
principlesandthegeometricpropertiesofthe3-dimensional
space.Theonlyqualitycriterioniswhetheramodellookslike
the known observations or can be tuned to coincide with
them. It is a free kingdom of modeling, admitting fancied
ghoststoexplainactualobservations,butanightmarewhen
looking for durable descriptions and durable explanations
thatwillnotbeoutdatedbyanynewobservationandthatare
prone to have their limits of validity checked theoretically.
Formyownwork,however,Ifoundsuchapproachesvery
stimulating. In hisgroundbreaking and beautifulwork, the
Figure 3. Crispin Van den Broeck (detail, 1577), with per-
mathematician Arthur Sherman (NIH, Bethesda, Maryland)
mission of the Royal Collection, London, chosen as the logo usesdynamicalsystemtechniquestostudythestrikingtwo
fortheinterdisciplinaryexocytosismodelinginitiative‘‘Getting phases character of secretion that is experimentally estab-
informationoutbeforetherapyin:Towardscuringdiabetesby lished:Uponstimulation,theinsulinreleaseofahealthycell
systemsbiologyofregulatedexocytosisinpancreaticb-cells’’. begins with a short peak of about five minutes length, fol-
lowed by a more steady and substantial release with a
alleviation of the specific failure. A few years ago, when
duration of about 25 minutes. In stressed or tired cells the
Ibroughttogetheraninternationalteamofmedicaldoctors
absenceofthefirstpeakisanearlyindicationofdegradation.
(from hospitals and industry), biologists, biophysicists,
Fornow,mathematicscannotfullyexplainthephenomenon,
powder chemists, electrical engineers, computer scientists
butitcanreproduceitinamulti-componentmodelandpoint
togetherwithmathematicians,wewerefrustratedaboutthe
topossiblycriticalparametersandthresholdvalues.Insucha
huge gap between the ubiquity of standardized symptom
way, Sherman’s work has indeed motivated many experi-
diagnosisandsymptomtreatmentontheonesideandhow
mentalandtheoreticalinvestigations(seeFig. 5).
little we knew about functioning and malfunctioning of
regulated exocytosis. Therefore, we chose the motto
The Phylogenetic Heritage
‘‘Getting information out before therapy in’’ with a corre-
Our insulin-producing b-cells are among the most differenti-
sponding logo (Fig. 3). Instead of short-term orientation
atedofallhumancells.Theyarecloselypackedwithazooof
toward immediately applicable and rapidly publishable
different types of organelles. Insulinlike peptides can be
results,weorientedtowardgoodoldfundamentalresearch.
detectedinourdistantinvertebrateancestorswhohavebeen
aroundformorethan600millionyears.Somethingresembling
pancreases with a kind of insulin-producing b-cells already
The Technological Push
existinthehagfish,whichhasbeenaroundformorethan500
Thisweknowfrommathematics:Readilyavailableelectronic
million years. So, any new observed process or measured
journals,largeuser-friendlydatabasesofmathematicalpre-
quantitymaybeirrelevant,arelic,aruinofhistoricaldevelop-
prints and reviews, efficient numerical software packages,
mentthathasnoimportanceanylonger.Ofcourse,thistypeof
anddesktopLaTeXeditingputusunderpressureasmathe-
confusionwasalsometinthehistoryofphysics.Howlongdidit
maticians. But all this is nothing compared to the immense
taketo assign to meteors and comets their placein ourcon-
technological pressure that cell research is subject to: with
ceptionofthesolarsystem,ortoremovePlutofromthelistof
each new generation of equipment, oceans of new data
planets?However,althoughtheruinsandrelicsideallysharpen
inundate us on quite different length scales. Rapidly
themindinsimpleresearchfieldssuchasphysicsandastron-
expanding technology-driven innovations deal with indi-
omy, they can be extremely confusing and even completely
vidualgenesintheDNA,withproteinsandwithelectricalcell
block medical research. Again and again one senses that we
membraneprocesses.Butalsothestructureandfunctionofa
mathematicianscomingfromtheoutsidearepossiblytooearly.
b-cellasawholecanbedescribedinmomentaryimages(by
Perhapswehadbetterwaitanother150or200years,untilthe
electrontomography)ordynamicsequences(bytrackingof
research has separated essential processes from nonessential
properlyprimednanoparticlesinlivingcells–adeveloping
processes,beforeweatlastcanbegintheseriouswork.
technology;seeFig. 4).
Heavy Preponderance of Ad-Hoc Perceptions Lack of Universality
Thereisnoshortageofheroicattemptsbysomescientiststo Whatstrikesmemostinmathematicalcellphysiologyisthe
bringorderandoverviewintothiswildjungleofrealdata. lackofanyuniversalityorscaleinvariance.Intheworldof
Most approaches, however, restrict themselves to ad-hoc physics,Maxwell’sequationsapplyforbothhigh-frequency
8
THEMATHEMATICALINTELLIGENCER
Figure 4. Three-colourconfocalmicroscopyofmagneticnanoparticles(MNPs)nondestructivelyenteredintoapancreaticb-cell
by low-frequency magneto-manipulation. Green: MNPs; red: cell membrane; blue: cell nucleus (courtesy E. Renstro¨m and
M. Koch). This crucial experiment delivered the proof-of-concept of the envisaged long-time (up to 10 minutes), precise and
localizablecapturingofintracellulardynamicsofpancreaticb-cells,namelybymanipulatingandtrackingMNPsinvivo–without
damaging the cell or blocking its function. (Note that live imaging is at the cost of the high resolution obtainable by electron
tomography offixed [frozen] objects; compare fig.2 andfig. 8).
radio waves and low-frequency voltage in power plants; mechanics.Thatmightbeconsideredawoundinphysics,but
the Navier-Stokes equations apply for both the continental itisauniquewound.Indiabetesresearch,wehavehundreds
atmosphericphenomenaandthewhirlingaroundashiphull. of such cracks and ditches where no one knows whether
In mathematical physics, we have concepts (such as fields thereisabridge,andifso,howitwouldbebuilt.
and ergodicity) relating point measurements with spatially
widespreadevents.Wenotedreluctance,evenresistance,in Volatility
thecellphysiologycommunitytosuchglobalconceptswhen Medical biology,as it is conducted today, is a hugeunder-
wepublishedanarticleincorporatingsuchideas.3Theywere takingwithmyriadarticlespublishedeveryyear.Notmanyof
perceivedasabstract,imaginary,speculative,andimmaterial. themwillbequoted2 yearslater.That’sprobablythereason
Of course there are cross connections in medicine whyakeyparameterforbibliometricresearchinformation,
between what we know on different length scales, about the impact factor, only examines the current references to
b-cellfunctionandourgeneticdata,themodeofoperationof papers that are not more than 2 years old. To be sure, the
single organs (such as the pancreas) and a body’s, an overall goal, the understanding of life and death, of health
organism’sbehaviorandtheperformanceofawholepopu- and illness, is long-term. But the angles of attack change
lation.Forexample,geneticdataarecollectedbyepidemi- constantly and appear frequently, dictated by some obser-
ologicalstudiesoflargepopulations,andthefeedbackiswell vational techniques that have just now come into use. The
studied between nutrient intake, liver and brain response, subjectseemstobecharacterizedbytheabsenceofestablished
andthesecretionsignaling.But–apartfromtheuniversality andgeneraltraditions.Aspracticedtoday,cellphysiologyis
of the applied statistical methods for parameter estimation ayoungsubjectthatisjustnowestablishingitself.Accidental
andhypothesistesting–allthemethodsusedarecloselytied discoveriesseemtoplayamajorrole.Werecognizethatfact
toaspecificbiologicallevel,aparticularlength,andatime alsofromphysics,where,forexample,thediscoveryofhigh-
scale.Weknowsuchahopelesssituationalsofrommathe- temperature superconductivity in conventional insulating
matical physics with the seeming incompatibility between ceramic materials by Bednorz and Mu¨ller in 1986 could
the mathematical theories of gravitation and quantum hardly be characterized as the result of deep theoretical
3D.Apushkinskayaetal.,Geometricandelectromagneticaspectsoffusionporemaking,inref.[1]infootnote1,pp.505-538.
9
(cid:2)2013SpringerScience+BusinessMediaNewYork,Volume35,Number1,2013
Figure 5. Extended six-pool compartment model, incorporating Ca-binding, of Y. D. Chen, S. Wang, and A. Sherman,
‘‘Identifying the targets of the amplifying pathway for insulin secretion in pancreatic beta cells by kinetic modeling of granule
exocytosis,’’ Biophys. J., 95/5 (Sept. 2008), 2226-2241. In a beautiful piece of analysis, the authors were able to reproduce the
typical two-phase insulin secretion of healthy b-cells by tuning the transition rates in their system of six coupled ordinary
differentialequations.Cellphysiologyandelectrontomographysupporttheauthors’claimofsixdistinguishablepoolsofinsulin-
packedvesicles.Whetherthevaluesofthefoundratescanbegivenabiophysicalinterpretationisstillanopenquestion.Itisalso
anopenquestionastowhetherthepresentauthor’sglobalelectromagneticpartmodel,ofthemakingofthefusionporeinjoint
work with D. Apushkinskaya, et al. in ref. [1] in footnote 1, pp. 507-538, supplements or contradicts the compartment model.
Figure drawn by H. Larsen, Roskilde. From B. Booss-Bavnbek, ‘‘Geometry and dynamics on nano scale: Towards a nano
geometry?’’Contemp.Math.584(2012),147-162,http://dx.doi.org/10.1090/conm/584/11600,p.159,reproducedwithpermission.
Original figure (cid:2)Amer.Math. Soc.
considerations. However, random breakthroughs certainly of mathematics is to check whether a random discovery
occurmoreofteninbiomedicine. deliverswhatitpromised.
Numericalproblemscanrapidlypileupwhen,forexam-
ple,onewantstosimulateafusionprocessofasimpleinsulin
Systems Thinking Versus Reductionism
vesicletotheplasmamembraneoftheb-cellthroughoutthe
Itgoeswithoutsayingthatastrictlyreductionistprogramis
process:thebendingoftheplasmamembraneintoadimple,
neededinmedicalresearch,ifthecurrentpackingofmedical
thecouplingofthevesicletothedimple,thecoalescenceof
ignoranceinad-hocassumptionsistobereplacedbyfalsifi-
vesicle and plasma membrane during the hemifusion, the
ablereferencestobasicphysicallaws.ButImustalsoacknowl-
formation of the fusion pore for emitting the insulin mole-
edgethatmostbodilyfunctionsandprocessesinvolvemany
cules, and dissolving the vesicle remains into the plasma
differentcellcomponents,neighboringcells,variousorgans,
membrane.Thereasonforthenumericalproblemsisthatwe
andthewholeorganisminaninteraction.Understandably,
areatamesoscale:thecharacteristiclengthsvaryfrom1 nm
theholisticsloganofsystemsbiologyhasbecomepopular,
forthelipidheads,to7 nmforthestrengthofthemembrane
andgreatexpectationsareattachedtoit.
bilayers of lipids – to 100-250 nm for the insulin vesicle
Bothprogramswillrevealexcitingnewfactsandrelations.
diameters.Thus,therelevantlengthsofregulatedexocytosis
Both approaches offer the mathematician rich working
considerablyexceedthelengthsthatchemistshavemastered
opportunities.Tome,themostpromisingdirectionissome-
usingMolecularDynamics(MD).Itisevenworsewiththe
whereinthemiddle:maybeafocusedsystemsbiologywill
time scale, because a simple b-cell responds to glucose
show its ability to touch the wall, knock a hole in it, and
stimulation by insulin secretion over 25-30 minutes. And
achieve a breakthrough. That has not happened yet. The
everythingisinthreedimensions(seeFig. 6).Thisrequires
hopeistodevelopamedicineandabiologythatsimplifiesin
thedevelopmentofspecialsoftwaretoaggregatebothspace
areductionistway,fearlesslyignoressomeprobablyrelevant
andtimeintervalstosomethingthatcurrentcomputerscan
aspects,andfocusesonalimitedrangeofprocesses–butin workwith.4
turnletsitselfholisticallyandequallyfearlesslybeconfronted
at a multitude of levels and a diversity of length and time
scalesallatonce.
The Dual Role of Mathematicians
Amathematiciancomingfromtheoutsidemustbehumblein
Mathematical Helping Hand
frontoftheimmensecalibrationandprogrammingworkthat
What,then,canamathematiciandointhisenvironment? underliessuch models.It’s hardnottosuccumbtothefas-
cinationofthe‘‘lively’’graphicaloutputofsuchsimulations.
The Daily Practice Respectfully and humbly, we should make our tool box
Justasinengineering,economics,oranywhereelse,alsoin available and fearlessly lend a handwhenneeded. But we
cell physiology the daily mathematical exercise consists of must not abandon our mathematical way of viewing, our
theestimationofsomeparameters,testingthesignificanceof acquiredcompetencetoinquireintothebasisforthemod-
some hypotheses, and designing compartment models for eling and the simulations. We must remain skeptical and
thedynamicsofcoupledquantitativevariables.Oftentherole questioneverythingbycross-checkingcalculations,insisting
4J.Shillcock,Probingcellulardynamicswithmesoscopicsimulations,inref.[1]infootnote1,pp.459-473.
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THEMATHEMATICALINTELLIGENCER