Maximum velocity of self-propulsion for an active segment

0.24 MB·English

by  Pierre Recho

Maximum velocity of self-propulsion for an active segment
P.Rechoa,b,L.Truskinovskya
aLMS,CNRS-UMR7649,EcolePolytechnique,RoutedeSaclay,91128Palaiseau,France
bPhysicochimieCurie,CNRS-UMR168,InstitutCurie,CentredeRecherche,26rued’UlmF-75248ParisCedex05,France
Abstract
5
Themotorpartofacrawlingeukaryoticcellcanberepresentedschematicallyasanactivecontinuumlayer. Themain
1
active processes in this layer are protrusion, originating from non-equilibrium polymerization of actin fibers, con-
0
2 traction,inducedbymyosinmolecularmotorsandattachmentduetoactivebondingoftrans-membraneproteinstoa
substrate.Allthreeactivemechanismsareregulatedbycomplexsignalingpathwaysinvolvingchemicalandmechani-
n
a calfeedbackloopswhosemicroscopicfunctioningisstillpoorlyunderstood.Inthissituation,itisinstructivetotakea
J reverseengineeringapproachandstudyaproblemoffindingthespatialorganizationofstandardactiveelementsinside
8 acrawlinglayerensuringanoptimalcost-performancetrade-off.Inthispaperweassumethat(intherangeofinterest)
2 theenergeticcostofself-propulsionisvelocityindependentandadopt,asanoptimalitycriterion,themaximizationof

theoverallvelocity.Wethenchooseaprototypicalsetting,formulatethecorrespondingvariationalproblemandobtain
]
h a set of bounds suggesting that radically different spatial distributions of adhesive complexes would be optimal de-
p pendingonthedomineeringactivemechanismofself-propulsion. Thus,forcontraction-dominatedmotility,adhesion
-
hastocooperatewith’pullers’whichlocalizeatthetrailingedgeofthecell,whileforprotrusion-dominatedmotility
o
i itmustconspirewith’pushers’concentratingattheleadingedgeofthecell. Bothtypesofcrawlingmechanismswere
b
observedexperimentally.
.
s
c
i
s
y
h 1. Introduction
p
[ Eukarioticcellsarespatiallyextendedactivebodiesthatcansteadilyself-propelinviscousenvironmentsatlow

Reynolds numbers [1, 2]. It has been understood [3, 4, 5] that in these conditions a combination of stationarity
1
v and linearity of friction leads to kinematic reversibility and that a symmetric (under time reversal) stroke cannot
4 produce self-propulsion. For Stokes swimmers, a variety of non-symmetric motility strategies have been proposed
9 and optimized using various efficiency criteria [6, 7, 8, 9, 10, 11, 12]; similar models for crawlers advancing on a
1
frictional background were considered in [13, 14, 15]. Most of the self-propulsion mechanisms proposed in these
7 papersarefullykinematicinthesensethatthetimedependenceoftheshapeofaswimmer/crawlerisprescribed.This
0
. impliesthatappropriatelychosenactuatorscanalwaysperformtherequiredinternalmovements. Incellstheroleof
1
suchactuatorsisplayedbyactiveagentsandinthispaperwefocusonthefactthattheirdynamics,whilebeingdriven
0
biochemically,mustbecompatiblewiththefundamentalbalancesofmassandmomentum.
5
1 In the context of eukaryotic cell motility, a prototypical scheme of self-propulsion includes protrusion through
: polymerization of actin filaments which is accompanied by dynamic assembly of focal adhesions; myosin-driven
v
contractionoftheactinnetworkwhichallowsthemotorparttoadvanceacargo,and,finally,detachmentofadhesive
i
X contactswithasimultaneousdepolymerizationofactinfibers[16]. Itisusuallyassumedthatactivepolymerization
r ensuringprotrusioncanbedescribedastheworkofspatiallydistributedpushers, generatingpositiveforcecouples,
a
while active contraction can be viewed as an outcome of the mechanical action of distributed pullers, responsible
for negative force couples [17, 18, 19, 20, 21]. The role of ATP in reversible adhesion of adhesive patches (focal
adhesions)isunderstoodratherpoorlyandtheyareusuallytreatedaspassiveviscousbinderswhosespatialdistribution
mayberegulatedactively[22].
Emailaddress:[email protected](L.Truskinovsky)
PreprintsubmittedtoMathematicsandMechanicsofSolids January29,2015
Since our knowledge of the mechanism controlling the transport and the intensity of active agents performing
protrusion,contractionandadhesionisratherlimited,weadoptinthispaperasemi-kinematicapproachandtreatthe
correspondingdistributionsasfunctionalcontrolparametersconstrainedbythefundamentalmechanicalbalances.We
thenposeavariationalproblemoffindingtheoptimaltemporalandspatialdistributionsoftheseparametersinsidea
crawlingcontinuumbody. Inviewofsomesuccessfulattemptstojustifysuchreverseengineeringapproach[23],we
anticipatethatouroptimalsolutionswillbeeventuallybackedbyanappropriateconstitutivetheory.
While the real organisms are expected to optimize some measure of a trade-off between the velocity of self-
propulsion and the corresponding energy expenditure, in this paper we make a simplifying assumption that the en-
ergetic cost of self-propulsion is fixed and use as optimality criterion the maximization of the overall velocity. We
are interested in steady translocation and assume that the internal distributions of mechanical parameters are com-
patible with the traveling wave ansatz. This simplifying assumption allows us to replace the optimization of the
crawling stroke in space and time by a purely spatial optimization of the internal distribution of active elements in
theco-movingcoordinatesystem. Intheinterestofanalytictransparencyweusethesimplest1Dmodelofactively
contractingcontinuum(subjectedtoviscousfrictionalforces)thathasbeenusedrepeatedlyinthecellmotilitystudies
[24,25,26,27,28,29,30,16,31]. Inasettingsimilartooursthedependenceofcellvelocityonthedistributionof
activestressesandadhesionpropertieswasstudiedin[18]wherebothcontractionandprotrusionwererepresentedby
activecouples. Giventhatprotrusionisusuallylocalizedattheleadingedgeofthecell,wemodeltheeffectofactive
polymerizationdifferentlybyusingStefantypeboundaryconditionsontheedgesofacrawlingsegmentthatfixthe
influxandtheoutflowofactin, seealso[32,33,21]. Forthegivenstrengthofprotrusion, weprescribetheaverage
levelofcontractileactivity,andthensearchfortheoptimalinternaldistributionofcontractileandadhesiveunits.
Our analysis of the ensuing variational problem demonstrates that radically different distributions of focal ad-
hesions are optimal depending on the domineering active mechanism of self-propulsion. Thus, for contraction-
dominated motility, focal adhesions have to cooperate with pullers which end up localizing at the trailing edge of
thecellwhileforprotrusion-dominatedmotilitytheymustconspirewithpusherswhichconcentrateinourmodelat
theleadingedgeofthecell. Bothtypesofcrawlingmechanismshavebeenobservedexperimentally.
Thepaperisorganizedasfollows. InSection2weformulatethemodel. Asimpleanalyticallytractablecaseof
contractiondominatedmotilityistreatedinSection3. InSection4weobtainanalyticallyupperandlowboundsfor
theself-propulsionvelocityinthegeneralcase. ThesameoptimizationproblemwasstudiednumericallyinSection
5. InSection6wepresentsomeevidencethatourlowerboundmayinfactcoincidewiththeoptimalsolution. The
lastSection7containsthediscussionofourresults.
2. Themodel
Following [29, 32], we consider a one-dimensional segment of viscous active gel representing the cell lamel-
lipodiumonafrictionalsubstrate. Thesegmenthastwofreeboundarieswhichweidentifyasthetrailingedgel (t)
−
andtheleadingedgel+(t). Theforcebalancecanbewrittenintheform
∂ σ=ξv (1)
x
whereσ(x,t)isthestressandv(x,t)isthevelocity.Weassumethatthefrictionalcoefficientmimickingthedistribution
offocaladhesions[33,34,32,35,36,37]isspaceandtimedependentξ(x,t) ≥ 0. Theconstitutivebehaviorofthe
gelismodeledbytheequation[29,32]
σ=η∂ v+χ, (2)
x
where the active pre-stress χ(x,t) ≥ 0 accounting for the presence of myosin molecular motors [38, 39, 40, 41, 42]
is also assumed to be a function of space and time. For simplicity we assume that the bulk viscosity coefficient
η > 0 is constant. The assumption of infinite compressibility in (2) allows us to decouple the transport of (actin)
densityρ(x,t)fromforcebalancemakingthemechanicalproblem’staticallydeterminate’.Themassbalanceequation
∂ρ+∂ (ρv)=0canthenbesolvedindependentlyafterthevelocityfieldisdetermined[43,44].
t x
We further assume that some internal mechanism (stiffness of the cell cortex [45, 46, 47, 48, 49, 50], osmotic
pressureactivelycontrolledbythechannelsandpumpsonthecellmembrane[51,52], etc.) maintainsagivensize
L0 = l+−l− ofthecell. Thereforethestressattheedgesmustbethesameσ(l−(t),t) = σ(l+(t),t) = σ0,whereσ0(t)
isthenanunknownfunction. TomodelactiveprotrusionweimposetwokinematicStefantypeboundaryconditions
2
characterizingtherateofactinpolymerizationv+ > 0anddepolymerizationv− > 0ontheboundariesofthemoving
segment[29,34,32,33,43]
l˙ =v +v(l (t),t). (3)
± ± ±
Forconsistency,theoverallmassbalancemustbealsorespectedonthemovingboundariesandwesetρ(l (t),t)v =
− −
ρ(l+(t),t)v+, which implies an instantaneous recycling of depolymerized actin from the trailing edge to the leading
edge,see[21,43]formoredetailonsuchclosureofthetreadmillingcycle. Whilethereisconsiderableexperimental
evidencethatactivepolymerizationisindeedlocalizedattheleadingedgeofacrawlingcell,thede-polymerization
maybespreadalongthelengthofthelamellipodium[33,32]. However,intheinterestofanalytictransparency,such
spreadingwillbeignoredinthisstudy,seethough[43].
The two functions χ and ξ can be interpreted as infinite dimensional controls parameters and found through an
optimizationprocedure. Evenintheabsenceofadetailedmicroscopicmodelgoverningtherearrangementofthese
agentswestillneedtosubjectthemtointegralconstraintsprescribingtheaveragenumberofadhesioncomplexes[42]
1 (cid:90) l+(t)
ξ(x,t)dx=ξ∗, (4)
L
0 l−(t)
whereξ∗ >0isagivenconstantand
1 (cid:90) l+(t)
χ(x,t)dx=χ∗, (5)
L
0 l−(t)
whereχ∗ > 0isanothergivenconstantrepresentingtheaveragenumberofcontractilemotors[53]. Itisclearfrom
(4,5) that since we prescribe the density of active agents, the performance of the self-propulsion machinery will be
proportionaltothelengthoftheactivesegment,sotheappropriatevelocityfunctionalmaybealsonormalizedbythe
totallength.
Tosimplifytheanalysiswefurtherassumethatthemotionoftheactivesegmentissteady[32,33]withunknown
velocity V = l˙− = l˙+ and that the unknown functions σ,v and the unknown controls ξ,χ depend exclusively on the
appropriately chosen co-moving coordinate u = (x− x −Vt)/L ∈ [−1/2,1/2] . Then in dimensionless variables
(cid:112) 0 0
σ:=σ/χ∗,x:= x/ η/ξ∗,t:=t/(η/χ∗),ξ :=ξ/ξ∗andχ:=χ/χ∗weobtaintheforcebalanceequation
(cid:32) (cid:33)
1 ∂ σ(u)
− ∂ u +σ(u)=g (u), (6)
L2 u g (u) 2
1
(cid:112)
whereL:= L / η/ξ∗. There-scaledcontrolfunctions
0
g (u)=ξ(Lu)≥0,g (u)=χ(Lu)≥0
1 2
mustsatisfytheconstraints
(cid:90) 1/2 (cid:90) 1/2
g (u)du= g (u)du=1. (7)
1 2
−1/2 −1/2
Theboundaryconditionstaketheform
(cid:40) σ(−1/2)=σ(1/2)
(cid:16) (cid:17) (8)
1 ∂uσ(1/2) − ∂uσ(−1/2) =−∆V
L2 g1(1/2) g1(−1/2)
where
∆V := (v+−v−)η.
χ∗L
0
ThedimensionlessvelocityofthesegmentperlengthV =V/Lcanbefoundfromtheformula
(cid:32) (cid:33)
1 ∂ σ(1/2) ∂ σ(−1/2)
V =V + u + u (9)
m 2L2 g (1/2) g (−1/2)
1 1
where
V := (v++v−)η.
m 2χ∗L
0
3
Now, if we assume that the two parameters (V ,∆V), characterizing actin treadmilling, are fixed we can pose the
m
optimization problem of finding the controls g (u), g (u) ensuring the maximization of the normalized velocity V.
1 2
This problem is nontrivial because the functional V{g ,g } is prescribed implicitly through the unknown solution
1 2
of the boundary value problem (6,8). To our advantage though this linear elliptic problem is classical and is well
understood,e.g. [54].
We observe that parameter V enters the expression for the velocity (9) in an additive way and does not affect
m
the solution of the optimization problem. The reason is that V characterizes a propulsion mode associated with
m
simpleaccretionofthematerialatthefrontanditssimultaneousremovalattherear;whenV (cid:44)0anaprioripolarity
m
is imposed and the problem of motility initiation disappears. In view of the complete decoupling of this mode of
self-propulsionfromourcontrols,inwhatfollowsweassumewithoutlossofgeneralitythatV =0.
m
The second parameter ∆V, also characterizing the protrusion strength, does not induce polarity. As we clarify
in the next Section, this parameter represents the mechanical action of pushers and the dependence of the crawling
velocityon∆V,whichisnowsensitivetobothcontrols,ismuchmoresubtlethaninthecaseofV .
m
3. Pushersandpullers
Toseethatparameter∆Vcharacterizesinoursettingtheactivityofpushers,considertheglobalbalanceofcouples
intheco-movingcoordinatesystem
(cid:90) 1/2 (cid:90) 1/2
L g (u)v(u)udu=−∆V+ g (u)du+σ . (10)
1 2 0
−1/2 −1/2
Heretheterminthelefthandsidecharacterizesthetotalmomentduetoexternal(frictional)forces. Thefirsttermin
thetherighthandside
T =∆V
is due to active protrusion, while the second term (cid:82)1/2 g (u)du = 1 is due to active contraction. The last term σ
−1/2 2 0
correspondstopassivereactionforcesresultingfromtheprescriptionofthelengthofthesegment.
Ourassumptionthat∆V >0meansthattheprotrusioncouplehasanegativesignshowingthatthecorresponding
forcedipolesactonthesurroundingmediumbypushingoutwardandcreatingnegativestress.Instead,thecontraction
couple has a positive sign because the contractile forces pull inward and the induced stresses are positive. We can
thereforeassociateprotrusionwiththepresenceofpushersandcontractionwiththeactivityofpullers[18,21]. We
canalso(tentatively)arguethatmotilityisprotrusion-dominatedwhenT > 1anditiscontraction-dominatedwhen
0<T <1. Thisassertionwillbeconfirmedlaterinthepaperbyrigorousanalysis.
To illustrate the different roles played in our motility mechanism by pushers and pullers, we present below an
analysisofatoymodelwhichanticipatesthemainconclusionsofthepaper. Wetemporarilyset∆V =0anddescribe
the distribution of pushers and pullers by the same function g (u) allowing it now to be both positive and negative.
2
Ourgoalistoshowthatforprotrusion-dominatedmotilitydrivenbypushers,itisbeneficialtocreatestrongadhesion
attheleadingedgewhileforcontraction-dominatedmotilitydrivenbypullers,itisthetrailingedgethathastoadhere
moststrongly.
Consideraspecialchoicesofcontrolfunctions,
g (u)=qδ(u−u )+(1−q)δ(u−u ),g (u)= pδ(u−u )−(1−p)δ(u−u ), (11)
1 1 2 2 3 4
whereδistheDiracdistribution,0 ≤ q,p ≤ 1and−1/2 ≤ u ,u ,u ,u ≤ 1/2. Thecontrolfunctiong (u)represents
1 2 3 4 1
twoadhesionsitesu = u andu = u whoselocationsandintensitiesmustbeoptimizedinrelationtotheprescribed
1 2
positionofasinglepullerplacedatu=u andcharacterizedbyapositivedipolemomentpandasinglepusherlocated
3
atu=u andcharacterizedbyanegativedipolemoment p−1. Theparameter pmeasurestherelativeimportanceof
4
contractioncomparingtoprotrusion.
Supposethat
u <u ,u <u (12)
3 1 2 4
4
whichensuresthatouractivesegmentmovesfromlefttorightandthatinthesegmenttheadhesionsitesarealways
outsidethelocationoftheactiveagents. Byusing(11)wecanexpressthevelocityofthesegmentas
(cid:32) (cid:33)
1 (1−2p)(u −u )c
V = 1+ 2 1 ,
2 u −u +1+(u −u )a
1 2 2 1
where,
1 1−2q
a= andc= .
1+q(1−q)(u −u )L2 1+q(1−q)(u −u )L2
2 1 2 1
Suppose for simplicity that our two adhesive complexes are placed symmetrically with respect to the center of the
segmentu =1−u . Wecanthenuseasingleparameter∆u=u +1/2=1/2−u toobtain,
1 2 1 2
 
V = 121+ 1+2(∆1u−q(21p−)(q1)−L22+q) 2∆u .
1−2∆u
At∆u=0,whenadhesivecomplexeslocalizeattheedgesofthesegment,thevelocityreachesitsmaximumvalue
V = 1(cid:2)1+(1−2p)(1−2q)(cid:3). (13)
2
Because of the imposed inequalities (12), we must necessarily have in this configuration u = −1/2 and u = 1/2
3 4
meaningthatpushersandpullersmustalsolocalizeattheedges. Noticehoweverthatinequalitiesu <u andu <u
3 1 2 4
arenecessaryforformula(13)toholdsotheapparentconclusionthatu = u = 1/2andu = u = −1/2isaresult
3 1 4 2
of an abuse of notation. In a theory where, pushers, pullers and adhesion complexes have a characteristic size of
dispersion,theadhesionclusterswillbeslightlyaheadofpullersandslightlybehindthepusherssothatactiveagents
takeadvantageofthefirmattachmenttoeitherpushorpull(seemoreonthissubjectbelow).
Fromoursimpleanalysisitfollowsthatifpullersdominate(p > 1/2),theoptimalwaytotuneadhesionistoset
q = 1andconcentrateadhesivecomplexesatthetrailingedgeofthemovingsegmentachievingmaximumvelocity
V = 1 when p = 1. If, instead, pushers dominate (p < 1/2), the optimal way to tune adhesion is to set q = 0 and
concentrate adhesive complexes at the leading edge. In this case, the maximal velocity is again V = 1 given that
p = 0. Inotherwords,tobeeffective,pullershavetoconcentrateonthetrailingedgeandensurestrongadhesionon
the leading edge: in this way pullers can inflict contraction that displaces the trailing edge which due to the length
constraint also propels the leading edge. On the contrary, pushers can take advantage of the firm attachment at the
trailing edge to push against it and propel the leading edge which in turn pulls the trailing edge due to the length
constraint.
4. Contractiondrivenmotility
We now return to the study of the optimization problem in the original formulation. The simplest analytically
transparentcaseiswhenprotrusionisdisabled∆V =0andmotilityisfullycontraction-driven.
Supposefirstthatg ≡ 1whichmeansthattheadhesioncomplexesaredistributeduniformly. Thenthevelocity
1
oftheactivesegmentcanbeexpressedasaquadrature
1 (cid:90) 1/2
V =− sinh(Lu)g (u)du. (14)
2sinh(L2) −1/2 2
Onecanseethatifthefunctiong (u)iseven,thenV =0. Thisresultcanbeinterpretedasananalogueofthefamous
2
Purcell’stheoremaboutthenecessityofnon-symmetricstrokesinStokesswimming[4,2]. Ifthedistributiong (u)is
2
non-symmetricand,forinstance,moremotorsareplacedattherearofthesegment,thevelocitywillbecomepositive.
Using the fact g (u) ≥ 0 we can also conclude from (14) that V ≤ 1/2. This upper bound is reached when all the
2
motorsarefullylocalizedattherearandg (u)=δ(u+1/2).
2
5
Now,considerthegeneralcasewhenthefocaladhesionsaredistributedinhomogeneously. Since(6)isaSturm-
Liouvilleproblem,itssolutioncanbewrittenas
(cid:90) 1/2
σ(u)=σ − G(u,s)(cid:2)g (s)−σ (cid:3)ds, (15)
0 2 0
−1/2
wheretheGreen’sfunctionG(u,s)canberepresentedtwoauxiliaryfunctionsh(u)and f(u)
G(u,s)= 1 (cid:2)h(u)f(s)(cid:49) +h(s)f(u)(cid:49) (cid:3), (16)
C [s<u] [u<s]
solvingthefollowingstandardboundaryvalueproblems[54]:
(cid:40) (cid:40)
(1h(cid:48))(cid:48) = L2h (1 f(cid:48))(cid:48) = L2f
h(−1/2g)1 =1,h(1/2)=1 , f(−1/2g)1=1, f(1/2)=−1 . (17)
In(16),C =(hf(cid:48)− fh(cid:48))/g isaconstantinvolvingtheWronskianofthetwoauxiliaryfunctionsh(u)and f(u)and(cid:49)
1
istheindicatorfunction. Wecannowwrite
1(cid:90) 1/2
V = f(u)(g (u)−gˆ )du, (18)
2 2
2
−1/2
whereweintroducedanewmeasureofinhomogeneityofcontraction:
(cid:82)1/2
h(u)g (u)du
gˆ = −1/2 2 .
2 (cid:82)1/2
h(u)du
−1/2
Ifbothfunctionsg (u)areeven,then f(u)isoddand,sincetheintegralofaproductofanoddandanevenfunctions
1,2
is equal to zero, we obtain that V = 0. The same result follows if we assume that contraction is homogeneous
g (u)=gˆ =1whiletheadhesiondistributiong (u)isarbitrary. Therefore,toensuremotilityat∆V =0,contraction
2 2 1
mustbeinhomogeneouswhileadhesionmaystillbeuniformprovidedcontractionisnoteven.
Tofindtheoptimaldistributionsg (u), g (u)weproceedintwosteps. WefirstshowthatV ≤ 1andthenfinda
1 2
configurationofcontrolsallowingthecelltoreachthisbound.
Noticethatwecanrewrite(18)intheform
(cid:32)(cid:90) (cid:90) (cid:33)
1
V = f(u)(g (u)−gˆ )du+ f(u)(g (u)−gˆ )du
2 2 2 2
2
S+ S−
where we defined the domains S− = {u/g2(u)≤gˆ2} and S+ = {u/g2(u)>gˆ2}. Applying the maximum principle to
(17)weobtainthat1≥h(u)≥0andh(u)≥ f(u)≥−h(u). Usingtheboundson f,wecanwrite
(cid:32)(cid:90) (cid:90) (cid:33)
1
V ≤ h(u)g (u)du+gˆ h(u)du .
2 2
2
S+ S−
Sincetheintegrandsarepositiveandh(u)≤1itfinallyfollowsthat
(cid:90) 1/2 (cid:90) 1/2
V ≤ h(u)g (u)du≤ g (u)du=1. (19)
2 2
−1/2 −1/2
Observethatinthecaseofahomogeneousdistributionofadhesiveclusters,thevelocitycouldreachonlyonehalfof
thismaximalvalue.
WenowshowthatthemaximalvelocityV =1canbereachedifbothcontrolsg (u)andg (u)arefullylocalized.
1 2
Takeθ>0andconsideraregularizeddistribution
1 θ
g (u;θ)= .
1 πθ2+(u−u )2
1
6
Forthischoiceofg (u)theauxiliaryfunctionsh(u)and f(u)canbewrittenexplicitlyintermofLegendrepolynomials.
1
Inthelimitθ→0andlim g (u;θ)=δ(u−u )weobtain
θ→0 1 1
(cid:40)
1ifu≤u
h(u)=1and f(u)= 1
−1ifu>u .
1
Byusingtheseexplicitexpressionsfortheauxiliaryfunctionswecanrewrite(18)intheform
1(cid:34)(cid:90) u1 (cid:90) 1/2 (cid:35)
V = g (u)du− g (u)du−2u . (20)
2 2 1
2
−1/2 u1
ifwenowsupposethatg (u)=δ(u−u )theexpressionforvelocityreducestoreducesto
2 2

V = 1 1−−22uu1ififuu2=<uu1
2 −1−21u if2u >1u
1 2 1
Itisnowclearthatthevelocityreachesitsmaximalvalueasu → −1/2whileu < u . Wecanthenformallywrite
1 2 1
u =u =−1/2andobtainthecontrolsg (u)=g (u)=δ(u+1/2)saturatingtheboundV =1.Notice,however,that
2 1 2 1
ifweassumedirectlyu = u → −1/2in(20),weobtainV = 1/2. Thisisinagreementwiththephysicalintuition
1 2
thattheanchoragepointmustbelocatedtotherightofthepullingforcedipole:inthiscasethepullingforcesadvance
therearedgeofthesegmentwithminimalslipping. Mathematically,weencounterherethecaseofnon-commutation
ofthelimitingproceduresu →−1/2andu →−1/2givingV =1onlyifthelimitsaretakenintheaboveorder.
2 1
To summarize, the optimization of the distribution of focal adhesions allows the contraction-dominated mecha-
nism of cell motility to reach the value of velocity which is twice as large as when the adhesion is uniform. This
meansthatinordertoimprovethemotilityperformancetheadhesionmachinerymustconspirewiththecontraction
machinerymakingsurethatboththemotorsandtheadhesivecentersarelocalizedatthetrailingedge. Interestingly,
exactlythistypeofcorrelationbetweenthestressescreatedbycontractionandthedistributionoffocaladhesionswas
observedinexperimentsandnumericalsimulations[55,56,57,58,59,22]. Thelocalizationofadhesioncomplexes
closetocelledges,wherecontractionisthestrongest,hasbeenalsoreportedoutsidethemotilitycontext[60,61,62].
5. Upperandlowerboundsforvelocityinthegeneralcase
We now turn to the general case where both contraction and protrusion are active. In particular, the protrusive
power will be characterized by the parameter ∆V = T > 0 which was assumed to be equal to zero in the previous
Section. Wecanthenwrite
V = 12(cid:82)(cid:82)−111///222hf((uu))dduuT +(cid:90)−11//22 f(u)(g2(u)−gˆ2)du. (21)
−1/2
Aswesee,thefirsttermintherighthandsideisassociatedwithprotrusion-based(orfilament-driven[63])motility
while,aswehavealreadyseen,thesecondtermisthecontributionduetocontraction-based(ormotor-driven[63])
motility. Wenoticethatifg (u)iseven,then f(u)isoddandh(u)iseven,leadingto
1
(cid:82)1/2
f(u)du
−1/2 =0.
(cid:82)1/2
h(u)du
−1/2
Ifg (u)isalsoeven,then
2
(cid:90) 1/2
f(u)(g (u)−gˆ )du=0.
2 2
−1/2
Inthiscasethevelocityofthesegmentisfullycontrolledbytheaccretionmechanismcharacterizedbytheparameter
V .
m
7
Considerfirstthecaseofprotrusion-drivenmotilitybyassumingthatcontractionishomogeneousg (u) ≡ 1and
2
thereforedoesnotcontributetotheoverallvelocity. Byusingagainthemaximumprincipleweobtaininequalities
(cid:82)1/2
f(u)du
−1≤ −1/2 ≤1,
(cid:82)1/2
h(u)du
−1/2
leadingtotheupperbound
T
V =V ≤ . (22)
p
2
Themaximumoftheprotrusivecontributiontovelocityisreachedwhen,g (u)=δ(u− 1),becauseinthiscaseh=1
1 2
and f = 1almosteverywhere. Observe,thattheoptimalsolutioninthecaseofprotrusion-drivenmotilityisinsome
senseoppositetothesolutiong (u)=δ(u+1/2)obtainedinthecaseofthecontraction-drivenmotility.
1
Basedon(19)and(22)wecannowarguethatinthecasewhenbothtreadmillingandcontractionarepresent,an
upperboundforvelocityis
T
V ≤ +1,
2
however,aswehavejustseen,inviewoftheincompatibilityofthecorrespondingoptimalcontrols,thisboundcannot
be reached. The optimal strategy for focal adhesions would then involve a compromise between the necessity to
localizeadhesionatthetrailingedgeinordertoassistthecontractionmechanismandthecompetingtrendtolocalize
adhesionattheleadingedgeinordertoimprovetheprotrusionpowerofthecell.
To obtain a lower bound for V we now consider a particular test function representing a weighted sum of our
competingoptimalcontrols,g (u)=qδ(u+1/2)+(1−q)δ(u−1/2). Wealsochoseg (u)=δ(u−u ),whereq∈[0,1]
1 2 0
andu ∈[−1/2,1/2]aretwoparameterstobeoptimized. Then,bysolving(17)weobtain,
0

f(u)= 1−2q1ififuu=∈]−−1/12/2,1/2[
 1+q(1−q)L−21ifu=1/2
and,

h(u)= 1 1ififuu=∈]−−1/12/2,1/2[
 1+q(1−q)L12 ifu=1/2,
whichleadstotheexpressionforthevelocity
T 1
V = (1−2q)+ (f(u )−(1−2q)h(u )).
0 0
2 2
Theoptimizationwithrespecttou givesu =−1/2and
0 0
T
V = −q(T −1).
2
Finally,optimizinginqweobtainthatifT <1,weneedtotakeq=0andifT >1,wegetq=1.Thisresult,illustrated
inFig1,suggeststhatthereisaswitchatT = 1betweenthecontraction-centeredoptimizationstrategy(q = 0)and
the protrusion-centered optimization strategy (q = 1). Notice that the switch takes place exactly when the negative
protrusiongeneratedcoupleT becomesequaltothepositivecontractilecoupleequalto1. AtaninterestingstateT =
1, thetwoactivemechanismsneutralizeeachotherandactivity-relateddipolesbecomeinvisiblebehindthepassive
contributionsinEq. (10): inthiscasetheoptimalpositionofactiveandadhesiveagentsbecomesindeterminate.
8
Figure1: Solidlines:Lowerboundontheoptimalvelocityofself-propulsionVasafunctionofthemeasureofthe(relative)protrusivestrength
T. Theoptimalstrategydependsonwhethercontraction(T <1)orprotrusion(T >1)dominates. Thedashedlinerepresentstheupperbound
obtainedbyformallysummingtheincompatibleupperboundsfortheprotrusionandcontractionbasedstrategies. Thedottedlinerepresentsa
sub-optimalstrategyobtainedundertheassumptionthatadhesionishomogeneous.Insetsillustratetheassociatedconfigurationsofcontrolsg1(u)
andg2(u).
6. Numericalsolutionoftheoptimizationproblem
To show that the low bound obtained in the previous Section is rather close to being optimal, here we solve the
optimizationproblemnumerically. Afinitedimensionalreductionoftheoriginalvariationalproblemisconstructed
byselecting N +2pointsu = i/(N +1)−1/2thatsubdividethesegment[-1/2,1/2]. Wethenlocalizeadhesionand
i
contractioninthesepointsbychoosingthecontrolfunctionsintheform
(cid:88)N (cid:88)N+1
g (u)= giδ(u−u),g (u)= giδ(u−u),
1 1 i 2 2 i
i=1 i=0
where (cid:80)N+1gi = (cid:80)N gi = 1 and gi ≥ 0,gi ≥ 0. In this way we also mimic the discrete nature of real adhesive
i=0 2 i=1 1 1 2
clustersandrealmyosinmotors,soinsomerespectsthediscreteproblemisevenmorerealisticthanthecontinuum
one.
Bysolvingtheauxiliarylinearellipticproblemsforthischoiceofcontrolsweobtainthatthefunctionsh(u)and
f(u)arepiece-wiseconstanth(u) = (cid:80)N A(cid:49) (u)and f(u) = (cid:80)N C(cid:49) (u). Thecoefficientswithi ∈ [2,N]
i=0 i [ui,ui+1[ i=0 i [ui,ui+1[
satisfytheequations
gi(A −A )+gigi−1A L2(u −u )=gi−1(A −A )
1 i−1 i−2 1 1 i−1 i i−1 1 i i−1
gi(C −C )+gigi−1C L2(u −u )=gi−1(C −C )
1 i−1 i−2 1 1 i−1 i i−1 1 i i−1
Theboundaryconditionsgive
A =1,A =1,C =1,C =−1.
0 N 0 N
WeusetheconventionsA−1 = A0,AN+1 = AN,C−1 =C0andCN+1 =CN toexpressthevelocityV intheform
V = 21T −(cid:88)Ni=+01gi2Ai+2Ai−1(cid:80)(cid:80)iiNN==00CAii((uuii++11−−uuii)) + 21(cid:88)Ni=+01gi2Ci+2Ci−1. (23)
9
Thefunction(23)wasoptimizednumericallywithrespecttoparametersgi andgi subjectedtotheappropriateequality
1 2
andinequalitytypeconstraints. Tofindtheglobalminimumweusedthemethodofsimulatedannealingwithinitial
guessescorrespondingtohomogeneousconfigurations. InFig.2,illustratingourresultsforN =100,onecanseethat
for T < 1 both optimal functions g (u) and g (u) are localized at the trailing edge. Instead, for T > 1 we observe
1 2
thatg (u)localizesattheleadingedgewhileg (u)localizesatthetrailingedge. Thevalueofthemaximalvelocity,
1 2
obtained numerically, is the same as in the bound up to an error proportional to the mesh size. Notice also that the
optimally spaced adhesion points are shifted from the locations of force dipoles by a mesh size. Thus, the solution
ofthenumerical(regularized)probleminthecontraction-dominatedregime(withT = 0.9),showninFig.2,clearly
distinguishes the optimal functions g (u) and g (u) that are both localized at the size of the mesh. The function
1 2
g (u) remains different from zero at one mesh size beyond the point where we already have g (u) = 0 (for positive
1 2
velocity). Intheprotrusion-dominatedregime(atT =1.1)showninFig.2,themeshsizeagainpreventslocalization
ofthefunctiong (u)exactlyattheleadingedgewhileg (u)localizesexactlyatthetrailingedge. Theseobservations
1 2
confirm that the regularized numerical solution is more realistic than its singular analytic prototype. Overall, our
numerical results are in full agreement with the analytic bounds which suggests that that these bounds are (nearly)
sharp.
Figure2: Numericalresultsfortheoptimizationofthefunction(23)fortwovaluesoftheprotrusivestrength,belowthethresholdT =0.9and
abovethethresholdT =1.1.Insetsmagnifythestructureoftheoptimalcontrolsneartheboundariesofthemovingsegment.ParameterL=10.
7. Localstabilityanalysis
In this Section we use a perturbation analysis to provide additional evidence that our lower bounds are close to
beingoptimal.
Inwhatfollowsweusethesuperscript◦toindicatetheunperturbedstateandsuperscript(cid:63)tomarkparameters
characterizingtheperturbedconfiguration. Weassumethatforallfunctionsφ(u)thefollowingexpansionholdsinthe
firstorder
φ(u)=φ◦(u)+(cid:15)φ(cid:63)(u),
where(cid:15) isasmallparameter. KeepingonlythefirstordertermintheexpressionforV,weobtain
V(cid:63) = 1(cid:34)(cid:82)1/2 g◦f(cid:63)+ f◦g(cid:63)− (cid:82)−11//22f◦ (cid:18)(cid:82)1/2 g◦h(cid:63)+h◦g(cid:63)(cid:19)
2 −1/2 2 2 (cid:82)1/2h◦ −1/2 2 2
+T−(cid:82)−11//22h◦g◦2 (cid:32)(cid:82)1/2 f(cid:63)−−1(cid:82)/−211//22h(cid:63) (cid:82)1/2 f◦(cid:33)(cid:35),
(cid:82)1/2h◦ −1/2 (cid:82)1/2h◦ −1/2
−1/2 −1/2
whereh(cid:63) = H(cid:63)(cid:48), f(cid:63) = F(cid:63)(cid:48)and
(cid:40) (cid:40)
H(cid:63)(cid:48)(cid:48)−L2g◦H(cid:63) = L2h◦g(cid:63) F(cid:63)(cid:48)(cid:48)−L2g◦F(cid:63) = L2F◦g(cid:63)
1 1 , 1 1 (24)
H(cid:63)(cid:48)(−1/2)= H(cid:63)(cid:48)(1/2)=0 F(cid:63)(cid:48)(−1/2)= F(cid:63)(cid:48)(1/2)=0.
Inviewofthesaturationoftheconstraintsbytheunperturbedsolution(cid:82)1/2 g◦ =(cid:82)1/2 g◦ =1,wemusthave
−1/2 2 −1/2 1
(cid:90) 1/2 (cid:90) 1/2
g(cid:63) = g(cid:63) =0.
2 1
−1/2 −1/2
10