Taxis Equations for Amoeboid Cells

4.8 MB·English

by  Radek Erban

SubmittedtotheJournalofMathematicalBiology
RadekErban·HansG.Othmer
6
0 Taxis Equations for Amoeboid Cells
0
2
February9,2008–(cid:13)c Springer-Verlag2008
n
a
J Abstract The classical macroscopic chemotaxis equations have previously been

9 derivedfromanindividual-baseddescriptionofthetacticresponseofcellsthatuse
2 a “run-and-tumble”strategyin responseto environmentalcues[17,18].Here we

derive macroscopicequationsfor the more complextype of behavioralresponse
]
B characteristicofcrawlingcells,whichdetectasignal,extractdirectionalinforma-
C tion from a scalar concentration field, and change their motile behavior accord-
ingly. We present severalmodels of increasingcomplexity for which the deriva-
.
o
tion ofpopulation-levelequationsis possible,andwe showhowexperimentally-
i
b measuredstatisticscanbeobtainedfromthetransportequationformalism.Wealso
- showthatamoeboidcellsthatdonotadapttoconstantsignalscanstillaggregate
q
[ insteadygradients,butnotinresponsetoperiodicwaves.Thisisincontrasttothe
caseofcellsthatusea“run-and-tumble”strategy,whereadaptationisessential.
1
v
8 1. Introduction
4
0 Motile organisms sense their environment and can respond to it by (i) directed
1 movementtoward or away from a signal, which is called taxis, (ii) by changing
0
their speed of movementand/orfrequencyof turning,whichis called kinesis, or
6
(iii)byacombinationofthese.Usuallytheseresponsesarebothcalledtaxes,and
0
/ weadoptthisconventionhere.Taxisinvolvesthreemajorcomponents:(i)anex-
o
ternalsignal,(ii)signaltransductionmachineryfortransducingtheexternalsignal
i
b into an internal signal, and (iii) internal components that respond to the trans-
-
ducedsignalandleadtochangesinthepatternofmotility.Inordertomoveaway
q
: from noxioussubstances (repellents) or toward food sources (attractants) organ-
v
isms must extract directional information from an extracellular scalar field, and
i
X there are two distinct strategies that are used to do this. A simple paradigm will
r illustratethese.
a
RadekErban:MathematicalInstitute,UniversityofOxford,24-29StGiles’,Oxford,OX1
3LB,UnitedKingdom,e-mail:[email protected]
Hans G. Othmer: School of Mathematics, 270A Vincent Hall, University of Minnesota,
Minneapolis,MN55455,USA,e-mail:[email protected]
Keywords:amoeboidcells,microscopicmodels, direction-sensing, aggregation, chemo-
taxisequation,velocityjumpprocess
2 R.ErbanandH.G.Othmer
Supposethat one is close enoughto a bakery to detect the odors,but cannot
seethebakery.Tofindit,onestrategyistousesensorsattheendofeacharmthat
measurethedifferenceinthesignalatthecurrentlocationandusethedifferenceto
decideonadirection.Clearlyhumansdonotusethisstrategy,butinstead,execute
the “bakery walk”, which is to take a sniff and judge the signal intensity at the
presentlocation,takeastepandanothersniff,comparethesignals,andfromthe
comparisondecideonthenextstep.
The first strategy is used by amoeboid cells (cells which move by crawling
through their environment),which have receptors on the cell membrane and are
large enough to detect typical differences in the signal over their body length.
Smallcellssuchasbacteriacannoteffectivelymakea“two-pointinspace”mea-
surementovertheirbodylength,andthereforetheyadoptthesecondstrategyand
measure the temporal variation in the signal as they move through the external
field.Ineithercase,animportantconsiderationinunderstandingpopulation-level
behavioris whether or not the individualmerely detects the signaland responds
to it, or whether the individualalters it as well, for example by consuming it or
byamplifyingitsoastorelaythesignal.Intheformercasethereisnofeedback
from the local density of individuals to the external field, but when the individ-
ualproducesordegradesthesignal,thereiscouplingbetweenthelocaldensityof
individuals and the intensity of the signal. The latter occurs, for example, when
individuals move toward a signal from neighboringcells and relay the signal as
well, as in the aggregation of the cellular slime mold Dictyostelium discoideum
(Dd).
Oneofthe best-characterizedsystemsthatadoptsthe “bakerywalk”strategy
is the flagellated bacterium E. coli, for which the signal transduction machinery
is well characterized[5]. E. colialternates between a moreor less linear motion
called a run and a highly erratic motion called tumbling, which produces little
translocationbutreorientsthecell. Runtimesaretypicallymuchlongerthanthe
tumbling time, and when bacteria move in a favorable direction (i.e., either in
the direction of foodstuffs or away from noxious substances), the run times are
increased further. Since these bacteria are too small to detect spatial differences
in the concentration of an attractant on the scale of a cell length, they choose a
new directionessentially atrandomat the endof a tumble,althoughithas some
biasin the directionofthe precedingrun [4]. The effectof alternatingthese two
modesofbehaviorand,inparticular,ofincreasingtherunlengthwhenmovingina
favorabledirection,isthatabacteriumexecutesathree-dimensionalrandomwalk
with drift in the favorable direction, when observed on a sufficiently long time
scale[3,30].Inaddition,thesebacteriaadapttoconstantsignallevelsandineffect
onlyaltertherunlengthinresponsetochangesinextracellularsignals.Modelsfor
signaltransductionandadaptationinthissystemhasbeendeveloped[46,2],anda
simplifiedversionofthefirstmodelhasbeenincorporatedintoapopulation-level
description of behavior [17,18]. The latter analysis shows how parameters that
characterizesignaltransductionandresponseinindividualcellsareembeddedin
the macroscopicsensitivity χ in the macroscopicchemotaxisequationdescribed
later.Havingthebacterialexampleinmind,wewillcallthe“bakerywalk”strategy
asa“run-and-tumble”strategyinwhatfollows.
TaxisEquationsforAmoeboidCells 3
Thedirectedmotionofamoeboidcells(e.g.Ddorleukocytes),whichiscru-
cial in embryonic development,wound repair, the immune response to bacterial
invasion, and tumor formation and metastasis, is much more complicated than
bacterial motion. Cells detect extracellular chemical and mechanical signals via
membranereceptors,and these trigger signaltransductioncascades thatproduce
intracellularsignals.Smalldifferencesintheextracellularsignaloverthecellare
amplifiedintolargeend-to-endintracellulardifferencesthatcontrolthemotilema-
chineryof the cell and therebydeterminethe spatial localizationof contactsites
with the substrate and the sites of force-generation needed to produce directed
motion [40,7]. Movementof Dd and other amoeboidcells involvesat least four
differentstages[36,43].(1)Cellsfirstextendlocalizedprotrusionsattheleading
edge, which take the form of lamellipodia,filopodia or pseudopodia.(2) Not all
protrusionsare persistent, in thatthey must anchorto the substrate or to another
cellinorderfortheremainderofthecelltofollow[44].Protrusionsarestabilized
byformationofadhesivecomplexes,whichserveassitesformolecularsignaling
andalso transmitmechanicalforceto the substrate.(3)Next, in fibroblastsacto-
myosinfilamentscontractatthefrontofthecellandpullthecellbodytowardthe
protrusion,whereasinDd,contractionisattherearandthecytoplasmissqueezed
forward.(4)Finallycellsdetachtheadhesivecontactsattherear,allowingthetail
ofthecelltofollowthemaincellbody.InDdtheadhesivecontactsarerelatively
weak and the cells move rapidly (∼ 20µm/min), whereas in fibroblasts they are
verystrongandcellsmoveslowly.Thecoordinationandcontrolofthiscomplex
processofdirectionsensing,amplificationofspatialdifferencesinthesignal,as-
semblyof the motile machinery,andcontrolof the attachmentto the substratum
involvesnumerousmoleculeswhose spatialdistributionservesto distinguishthe
frontfromtherearofthecell,andwhosetemporalexpressionistightlycontrolled.
Inaddition,Ddcellsadapttothemeanextracellularsignallevel[40].
Dd is a widely-used model system for studying signal transduction, chemo-
taxis, and cell motility. Dd uses cAMP as a messenger for signaling initiated by
pacemaker cells to control cell movement in various stages of development(re-
viewed in [39]). In the absence of cAMP stimuli Dd cells extend pseudopods
in more-or-less random directions, although not strictly so since formation of a
pseudopodinhibitsformationofanotheronenearbyforsometime.Aggregation-
competentcellsrespondtocAMPstimulibysuppressingexistingpseudopodsand
roundingup(the“cringeresponse”),whichoccurswithinabout20secsandlasts
about30secs[8].UnderuniformelevationoftheambientcAMPthisisfollowed
by extensionof pseudopodsin variousdirectionsand an increase in the motility
[53,54].However,onepseudopodusuallydominates,evenunderuniformstimu-
lation.AlocalizedapplicationofcAMPelicitsthe“cringeresponse”followedby
alocalizedextensionofapseudopodnearthepointofapplicationofthestimulus
[47].Howthecelldeterminesthedirectioninwhichthesignalislargest,andhow
itorganizesthemotilemachinerytopolarizeandmoveinthatdirection,arema-
jorquestionsfromboththeexperimentalandtheoreticalviewpoint.SincecAMP
receptors remain uniformly distributed around the cell membrane during a tac-
ticresponse,receptorlocalizationoraggregationisnotpartoftheresponse[27].
Well-polarized cells are able to detect and respond to chemoattractantgradients
4 R.ErbanandH.G.Othmer
withaslittle asa 2%concentrationdifferencebetweentheanteriorandposterior
ofthecell[34].Directionalchangesofashallowgradientinducepolarizedcellsto
turnonatimescaleof2-3seconds[23],whereaslargechangesleadtolarge-scale
disassemblyofmotilecomponentsandcreationofanew“leadingedge”directed
towardthestimulus[22].Polarityislabileincellsstarvedforshortperiodsinthat
cellscanrapidlychangetheirleadingedgewhenthestimulusismoved[47].
There are a number of models for how cells extract directional information
from the cAMP field. Fisher et al. [19] suggest that directional information is
obtainedbytheextensionofpseudopodsbearingcAMPreceptorsandthatsensing
thetemporalchangeexperiencedbyareceptorisequivalenttosensingthespatial
gradient.However,Ddcellscontaina cAMP-degradingenzymeontheirsurface,
and it has been shown that as a result, the cAMP concentration increases in all
directionsnormalto the cellsurface[10]. Furthermore,morerecentexperiments
show that cells in a steady gradient can polarize in the direction of the gradient
without extendingpseudopods[40]. Thus cells must rely entirely on differences
in the signal across the cell body for orientation. Moreover, the timing between
differentcomponentsoftheresponseiscritical,becauseacellmustdecidehowto
movebeforeitbeginstorelaythesignal.AnalysisofamodelforthecAMPrelay
pathwayshowsthatacellexperiencesasignificantdifferenceinthefront-to-back
ratioofcAMPwhenaneighboringcellbeginstosignal[10],whichdemonstrates
thatsufficientend-to-enddifferencesforreliableorientationcanbegeneratedfor
typicalextracellularsignals.Anactivator-inhibitormodelforanamplificationstep
inchemotacticallysensitivecellswasalsopostulated[35].Amplificationofsmall
externaldifferencesinvolvesaTuringinstabilityintheactivator-inhibitorsystem,
coupledtoa slowerinactivatorthatsuppressestheprimaryactivation.While this
modelreproducessomeoftheobservedbehavior,thereisnobiochemicalbasisfor
it;itispurelyhypotheticalandomitssomeofthemajorknownprocesses.Amodel
that takes into accountsome of the known biochemicalsteps has been proposed
morerecently[31].
Theobjectiveofthispaperisto deriveequationsforthepopulation-levelbe-
haviorofamoeboidcellssuchasDdorleukocytesthatincorporatedetailsaboutthe
individual-basedresponsetosignals.Wepresentseveralmodelswiththeincreased
complexityforwhichthederivationofpopulation-levelequationsispossible.We
show how experimentally-measuredstatistics can be obtainedfromthe transport
equation formalism. The paper is organizedas follows. We discuss the classical
chemotaxis description and summarize the state of the art of the derivation of
macroscopicequationsandpopulation-levelstatisticsfromindividual-basedmod-
elsintheremainderofthissection.InSection2,weestablishthegeneralsetupfor
modelsofamoeboidcellsandwepresentindividual-basedmodelswhichcapture
the essentialbehavioralresponsesof eukaryoticcells. InSection 3we derivethe
macroscopicmomentequationsfromthemicroscopicmodelandthedependence
ofthemeanspeedonthesignalstrengthisstudied.Finally,weprovideconclusions
andthediscussionofthepresentedapproachesinSection4.
TaxisEquationsforAmoeboidCells 5
1.1. Macroscopicdescriptionsofchemotaxis
The simplest description of cell movementin the presence of both diffusiveand
tacticcomponentsresultsbypostulatingthatthefluxofcellsjisgivenby
j=−D∇n+nu , (1)
c
wherenisthedensityofcells,u isthemacroscopicchemotacticvelocityandD
c
isthedifusionconstant.Thetaxisispositiveornegativeaccordingasu isparallel
c
oranti-paralleltothedirectionofincreaseofthechemotacticsubstanceS.Keller
andSegel[28]postulatedthatthechemotacticvelocityisgivenbyu =χ(S)∇S
c
andthen(1)canbewrittenas
j= −D∇n+nχ(S)∇S (2)
whereχ(S)iscalledthechemotacticsensitivity.Intheabsenceofcelldivisionor
deaththeresultingconservationequationforthecelldensityn(x,t)is
∂n
=∇·(D∇n−nχ(S)∇S) (3)
∂t
and this is called a classical chemotaxis equation. Unless the distribution of the
chemotactic substance is fixed, (3) is coupled to an evolution equation for this
substance,andperhapsothergoverningvariables.
Other phenomenologicalapproachesto the derivationof the chemotactic ve-
locityhavebeentaken.Forexample,byapproachingtaxisfromamechanicalpoint
ofview,PateandOthmer[41]derivedthevelocityintermsofforcesexertedbyan
amoeboidcell.StartingfromNewton’slaw,neglectinginertialeffects,andassum-
ingthatthemotiveforceexertedbyacellisafunctionoftheattractantconcentra-
tion,theyshowedhowthechemotacticsensitivityisrelatedtotherateofchangeof
theforcewithattractantconcentration.Inthisformulationthedependenceofthe
fluxonthegradientoftheattractantarisesfromthedifferenceintheforceexerted
indifferentdirectionsduetodifferentattractantconcentrations.Experimentalsup-
port for this comes from work of [51], who show that as many pseudopods are
produceddown-gradientasup,butthoseup-gradientaremoresuccessfulingen-
eratingcellmovement.Weshalluseaversionofthemechanicalapproachtotaxis
inamodeldescribedinthefollowingsection.
Thefirstderivationthatdirectlyrelatesthechemotacticvelocitytoproperties
of individual cells is due to Patlak [42], who used kinetic theory arguments to
expressu intermsofaveragesofthevelocitiesandruntimesofindividualcells.
c
ThisapproachwasextendedbyAlt[1],whoshowedthatfora classofreceptor-
basedmodelsthefluxisapproximatelygivenby(2).Theseapproachesarebased
onvelocity-jumpprocesses,whichleadtotransportequationsoftheform
∂
′ ′ ′
p(x,v,t)+v·∇p(x,v,t) =−λp(x,v,t)+λ T(v,v)p(x,v,t)dv. (4)
∂t Z
V
where p(x,v,t) is the density of cells at position x ∈ Ω ⊂ Rn, moving with
velocity v ∈ V ⊂ Rn at time t ≥ 0, λ is the turning rate and kernel T(v,v′)
6 R.ErbanandH.G.Othmer
givestheprobabilityofachangeinvelocityfromv′ tov, giventhatareorienta-
tionoccurs[37].Externalsignalsentereitherthroughadirecteffectontheturning
rateλandtheturningkernelT,orindirectlyviainternalvariablesthatreflectthe
externalsignalandinturninfluenceλand/orT.Thefirstcaseariseswhenexper-
imental results are used to directly estimate parametersin the equation [20], but
thelatterapproachismorefundamental.Thereductionof(4)tothemacroscopic
chemotaxisequationsforthefirstcaseisdonein[24,38]and[6].
Somestatisticsofthedensitydistributioninthefirstcase,whereintheexternal
fieldmodifiestheturningkernelorturningratedirectly,caneasilybederivedand
used to interpret experimental data. To outline the procedure, we consider two-
dimensionalmotionofamoeboidcellsinaconstantchemotacticgradientdirected
alongthepositivex1axisoftheplane,i.e.
∇S =k∇Ske1, wherewedenoted e1 =[1,0]. (5)
Moreover, we assume that the gradient only influences the turn angle distribu-
tion T;details of the procedureare given in [37]. We assume for simplicity that
the individualsmovewith a constantspeeds. i.e.a velocityofan individualcan
be expressed as v(φ) ≡ s[cos(φ),sin(φ)] where φ ∈ [0,2π). We assume that
T(v,v′) ≡ T(φ,φ′)isthesumofasymmetricprobabilitydistributionh(φ)and
abiastermk(φ)thatresultsfromthegradientofthechemotacticsubstance.Since
thegradientisdirectedalongthepositivex1 axis,weassumethatthebiasissym-
metric about φ = 0 and takes its maximum there. Thus we write T(φ,φ′) =
h(φ−φ′)+k(φ)wherehandkarenormalizedasfollows.
2π 2π
h(φ)dφ=1 k(φ)dφ=0 (6)
Z0 Z0
Let p(x,φ,t) be the density of cells at position x ∈ R2, moving with velocity
v(φ) ≡ s[cos(φ),sin(φ)],φ ∈ [0,2π),attimet ≥ 0.Thestatisticsofinterestare
themeanlocationofcellsX(t),theirmeansquareddisplacementD2(t),andtheir
meanvelocityV(t),whicharedefinedasfollows.
1 2π
X(t)= xp(x,φ,t) dφdx,
N0 ZR2Z0
1 2π
D2(t)= kxk2 p(x,φ,t) dφdx,
N0 ZR2Z0
1 2π
V(t)= v(φ)p(x,φ,t) dφdx,
N0 ZR2Z0
1 2π
B(t)= (x·v(φ))p(x,φ,t) dφdx,
N0 ZR2Z0
whereN0 isthetotalnumberofindividualspresentandB(t)isanauxiliaryvari-
able that is neededin the analysis. Two furtherquantitiesthatarise naturallyare
the taxis coefficientχ, which is analogousto the chemotactic sensitivity defined
TaxisEquationsforAmoeboidCells 7
earlierbecauseitmeasurestheresponsetoadirectionalsignal,andthepersistence
indexψ .Thesearedefinedas
d
2π π
χ≡ k(φ)cosφdφ and ψ =2 h(φ)cosφdφ. (7)
d
Z0 Z0
The persistence index measures the tendencyof a cell to continuein the current
direction.Sincewehaveassumedthatthespeedisconstant,wemustalsoassume
thatχandψ satisfytherelationχ<1−ψ ,forotherwisetheformerassumption
d d
isviolated(cf.(10)).
Onecannowshow,bytakingmomentsof(4),using(6)andsymmetriesofh
andk,thatthemomentssatisfythefollowingevolutionequations[37].
dX dV
=V =−λ0V+λχse1 (8)
dt dt
dD2 dB
=2B =s2−λ0B+λχsX1 (9)
dt dt
whereλ0 ≡λ(1−ψd).Thesolutionof(8)subjecttozeroinitialdatais
1
X(t)=sCI t− (1−e−λ0t) e1, V(t)=sCI(1−e−λ0t)e1 (10)
(cid:18) λ0 (cid:19)
whereC ≡ χ/(1−ψ ) is sometimescalled the chemotropismindex.Thusthe
I d
mean velocity of cell movement is parallel to the direction of the chemotactic
gradient and approaches V∞ = sCIe1 as t → ∞. Thus the asymptotic mean
speedisthecellspeeddecreasedbythefactorC .
I
A measure of the fluctuations of the cell path around the expected value is
providedbythemeansquaredeviation,whichisdefinedas
1 2π
σ2(t)= kx−X(t)k2 p(x,φ,t)dφdx=D2(t)−kX(t)k2 . (11)
N0 ZR2Z0
Using(8)–(9),onealsofindsadifferentialequationforσ2.Solvingthisequation,
wefind
2s2 1 5
σ2 ∼ (1−C2)t+ C2−1 as t→∞
λ0 (cid:26) I λ0 (cid:18)2 I (cid:19)(cid:27)
andfromthisonecanextractthediffusioncoefficientas
2s2
D = (1−C2).
λ0 I
Thereforeiftheeffectofanexternalgradientcanbequantifiedexperimentallyand
representedasthedistributionk,themacroscopicdiffusioncoefficient,thepersis-
tenceindex,andthechemotacticsensitivitycanbecomputedfrommeasurements
of the mean displacement,the asymptoticspeed and the mean-squareddisplace-
ment.
However,it is not as straightforwardto derive directly the macroscopic evo-
lution equations based on detailed models of signal transduction and response.
8 R.ErbanandH.G.Othmer
Suppose that the internal dynamics that describe signal detection, transduction,
processingandresponsearedescribedbythesystem
dy
=f(y,S) (12)
dt
where y ∈ Rm is the vector of internal variables and S is the chemotactic sub-
stance (S is extracellular cAMP for Dd aggregation). Models that describe the
cAMPtransductionpathwayexist[33,48,49],butfordescribingchemotaxisone
would have to formulatea more detailedmodel. The formof this system can be
verygeneralbutitshouldalwayshavethe“adaptive”propertythatthesteady-state
value(correspondingtotheconstantstimulus)oftheappropriateinternalvariable
(the“responseregulator”)isindependentoftheabsolutevalueofthestimulus,and
thatthesteadystateisgloballyattractingwithrespecttothepositiveconeofRm.
We showedearlierthatfornon-interactingwalkerstheinternaldynamicscan
be incorporatedin the transportequationasfollows[17]. Letp(x,v,y,t) bethe
densityofindividualsina(2N +m)−dimensionalphasespacewithcoordinates
[x,v,y],wherex ∈RN isthepositionofacell,v ∈V ⊂ RN isitsvelocityand
y ∈ Y ⊂ Rm isitsinternalstate,whichevolvesaccordingto(12).Theevolution
ofpisgovernedbythetransportequation
∂p
′ ′ ′
+∇ ·vp+∇ ·fp=−λ(y)p+ λ(y)T(v,v,y)p(x,v ,y,t)dv (13)
x y
∂t Z
V
where,asbefore,weassumethattherandomvelocitychangesaretheresultofa
Poisson processofintensityλ(y). ThekernelT(v,v′,y)givesthe probabilityof
achangeinvelocityfromv′ tov,giventhatareorientationoccurs.ThekernelT
is non-negativeand satisfies the normalization condition T(v,v′,y)dv = 1.
V
Toconnectthiswiththechemotaxisequation(3),wehaveRtoderiveanevolution
equationforthemacroscopicdensityofindividuals
n(x,t)= p(x,v,y,t)dvdy. (14)
Z Z
Y V
The problemturnsoutto be tractable forsystems thatexecute“run-and-tumble”
motion,suchasE.coli.Toillustratethis,assumeforsimplicitythatthemotionis
restricted to 1D, the signal is time-independent,the speed s is constant, and the
turningphaseisneglected;thegeneralcasesaretreatedelsewhere[17,18].Letp+
(resp.p−)bethedensityofindividualsmovingtotheright(resp.left).Then(13)
leadstoatelegraphprocessdescribedbythehyperbolicsystem
∂p+ ∂p+ m ∂
+s + f (y,S)p+ =λ(y) −p++p− , (15)
i
∂t ∂x ∂y
Xi=1 i (cid:2) (cid:3) (cid:2) (cid:3)
− − m
∂p ∂p ∂
−s + f (y,S)p− =λ(y) p+−p− . (16)
i
∂t ∂x ∂y
Xi=1 i (cid:2) (cid:3) (cid:2) (cid:3)
TaxisEquationsforAmoeboidCells 9
Theessentialcomponentsoftheinternaldynamicsinthebacterialcontextarefast
excitation,followedbysloweradaptationandreturntothebasalturningrate,and
theseaspectsarecapturedinthesystem[39]
dy1 g(S(x))−(y1+y2) dy2 g(S(x))−y2
= and = . (17)
dt τ dt τ
e a
Hereg encodesthefirststepofsignaltransduction,S isthechemoattractant,and
τ andτ aretimeconstantsforexcitationandadaptation,respectively.Thecom-
e a
ponent y1 adapts perfectly to constant stimuli, i.e., the steady state response is
independent of the magnitude of the stimulus S. To obtain a macroscopic limit
equationforthetotaldensityn(x,t)weincorporatethevariablesy intothestate
i
andderiveasystemoffourmomentequationsforvariousdensitiesandfluxes[17].
Assumingthattheturningratehastheformλ(y) = λ0−by1,forλ0 > 0,b > 0,
we show that this system reduces to the classical chemotaxis equation for large
times
∂n ∂ s2 ∂n bs2τ g′(S(x))
a ′
= − S (x)n (18)
∂t ∂x(cid:18)2λ0 ∂x (cid:20)λ0(1+2λ0τa)(1+2λ0τe)(cid:21) (cid:19)
where the chemotactic sensitivity is given explicitly in terms of parameters that
characterize signal transduction and response. We have only used the simplified
dynamics (17) to obtain the macroscopic chemotactic sensitivity, but this model
capturestheessentialaspectsforbacterialtaxis[46,17].Anopenproblemishow
oneextractstheelementaryprocessesofexcitationandadaptationfromacomplex
networkofthetypeusedforsignaltransductioninE.coli.Finally,letusnotethat
theglobalexistenceresultsfor(13)whichiscoupledwiththeevolutionequation
fortheextracellularsignalwererecentlygivenin[14].
Equation (18) was derived for cells such as bacteria, that use the “run-and-
tumble”strategy,andourobjectiveinthispaperistoattemptasimilarreduction
of the transport equation to a chemotaxis equation for more complex amoeboid
eukaryoticcells.Inthefollowingsectionweintroducethegeneralsetupforstudy-
ing amoeboidtaxis. Then we study several “caricature” or “cartoon”models for
amoeboidchemotaxiswiththeobjectiveofderivingmacroscopicpopulation-level
equationsin eachcase. We start with a modelwhichcan captureinterestingfea-
turesofeukaryoticmotilitywithoutintroducingadditionalinternalstatevariables,
andthenaddinternalstatevariablestothemodel.
2. Amoeboidtaxiswithinternalvariables
Afundamentalassumptionintheuseofvelocity-jumpprocesses[37]todescribe
cellmotionisthatthejumpsareinstantaneous,andthereforetheforcesareDirac
distributions.Thisapproximatesthecaseinwhichverylargeforcesactoververy
shorttime intervals,andevenif one incorporatesa restingor tumblingphase, as
was done in [38], the macroscopic description of motion is unchanged. This is
appropriatefortheanalysisofbacterialmotion(andothersystemsthatusea“run-
and-tumble”strategy),assummarizedabove,sincetheeffectoftheexternalsignal
10 R.ErbanandH.G.Othmer
is to changethe rotationalbehaviorof the flagella, and not, so far as it is under-
stood, to affect the force generation mechanism itself. However, the situation is
verydifferentwhen analyzingthe movementof crawlingcells, for here the con-
troloftheforce-generationmachineryisanessentialcomponentoftheresponse.
WhileamoeboidcellssuchasDdextendpseudopods“randomly”intheabsenceof
signals,thedirectionofextensionistightlycontrolledinthepresenceofadirected
external signal, and the direction in which forces are exerted on the substrate is
controlledviathelocationofcontactswiththesubstrate.Thereforeitisappropri-
atetoincorporatetheforce-generationmachineryaspartoftheinternalstate,and
asafirststepwecondensethisallintoadescriptionofhowtheforceexertedbya
cellonitssurroundings(andvice-versa)dependsontheexternalsignal.Inreality
amoeboidcells are also highly deformable,and a completetheoreticaltreatment
oftaxisatthesinglecelllevelhastotakethisintoaccount.Thisiscurrentlyunder
investigationbutwillnotbepursuedhere;insteadweonlydescribethemotionof
thecentroidofthecell.However,thefollowingframeworkissufficientlygeneral
toallowdistributedinternalvariableswithinacell.
Hereafterwe use y as it appearsin (19) to denote the chemicalvariablesin-
volvedinsignaltransduction,controlofactinpolymerization,etc,andwedenote
theforceperunitmassonthecentroidofacellbyF(x,v,y).Thereforetheinter-
nalstateequationsaregivenby
dy
=G(y,S) (19)
dt
andthevelocityevolvesaccordingto
dv
=F(x,v,y). (20)
dt
HereG :Y×S→YisingeneralamappingbetweensuitableBanachspacesand
F : RN ×RN ×Y → RN whereN = 1,2,or3isthedimensionofthephysi-
calspace.Thisgeneralityisneededbecausethevariableycanincludequantities
that depend on the location in the cell or on the membrane, and which may, for
example,satisfyareaction-diffusionequationoranotherevolutionequation.
The cell is therefore described by the position and velocity of its centroid,
and the internal state y ∈ Y. In some importantcases described later there is a
projectionP :Y→Z⊂YfromYontoasuitablefinite-dimensionalsubspaceZ,
obtainedforexamplebyconsideringthefirstfewmodesinasuitablebasisforY,
suchthat
P(G(y,S))=G(z,S) and F(x,v,y)=F(x,v,z), where z≡Py.
(21)
HereG(·,S):Z→ZandF(·,·,·):RN×RN×Z→RN aremappingsbetween
finite-dimensionalspaces.ThefirstequalitydefinesthefunctionG,thoughitmay
ofcoursebedifficulttofindwhenGisnonlinear.ThefunctionFisexplicitlygiven
bythesecondequalitywhenthereductionispossible.
GivenasuitablechoiceoftheprojectionP,wecanreducetheinfinite-dimen-
sionalsystem(19)–(20)tothefollowingsetofordinarydifferentialequationsin