Force Modulating Dynamic Disorder: Physical Theory of Catch-slip bond Transitions in Receptor-Ligand Forced Dissociation Experiments

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by  Fei Liu

Force Modulating Dynamic Disorder: Physical Theory of Catch-slip bond
Transitions in Receptor-Ligand Forced Dissociation Experiments
Fei Liu1∗ and Zhong-can Ou-Yang1,2
1Centerfor Advanced Study,Tsinghua University, Beijing, China and
2Instituteof Theoretical Physics, The Chinese Academy of Sciences, P.O.Box 2735 Beijing 100080, China∗
(Dated: February 9, 2008)
Recentlyexperimentsshowedthatsomeadhesivereceptor-ligandcomplexesincreasetheirlifetimes
6 whentheyarestretchedbymechanicalforce,whiletheforceincreasebeyondsomethresholdstheir
0 lifetimesdecrease. Severalspecificchemicalkineticmodelshavebeendevelopedtoexplainthein-
0 triguingtransitionsfromthe“catch-bonds”tothe“slip-bonds”. Inthisworkwesuggestthatthe
2 counterintuitiveforceddissociationofthecomplexesisatypicalrateprocesswithdynamicdisor-
der. Anuniformone-dimensionforcemodulatingAgmon-Hopfieldmodelisusedtoquantitatively

n describethetransitionsobserved inthesinglebondP-selctinglycoproteinligand1(PSGL-1)−P-
a selectinforceddissociationexperiments,whichwererespectivelycarriedoutontheconstantforce
J [Marshall,et al., (2003) Nature 423,190-193] andthe forcesteady- or jump-ramp[Evans et al.,
2 (2004)Proc. Natl. Acad. Sci. USA98,11281-11286]modes. Ourcalculationshowsthatthenovel
1 catch-slip bond transition arises from a competition of the two components of external applied
forcealongthe dissociation reaction coordinate andthe complex conformational coordinate: the

formeracceleratesthedissociationbyloweringtheheightoftheenergybarrierbetweenthebound
]
B andfreestates (slip),whilethelaterstabilizesthecomplexbydraggingthesystemtothehigher
barrierheight(catch).
C
.
o
bi Adhesive receptor-ligand complexes with unique ki- where ko0ff is the intrinsic dissociation rate constant in
- netic and mechanical properties paly key roles in cell the absence of force, ξ‡ is the distance from the bound
q
aggregation, adhesion and other life’s functions in cells. state to the energy barrier, f is a projection of exter-
[
A well studied example is the receptors in selectin fam- nal applied force along the dissociation coordinate, kB
1 ily which comprises E-, L- and P-selectin interacting the Boltzmann’s constant, and T is absolute tempera-
v and forming “bonds” with their glycoprotein ligands. ture. The validity of the model has been demonstrated
7
These bonds are primarily responsible for the tethering in experiments (11; 12). Although later at least four
1
and rolling of leukocytes on inflamed endothelium un- models have been put forward to explain and under-
0
der shear stress (1; 2). In particular, in the past two stand various receptor-ligand forced dissociation exper-
1
0 years great experimental efforts (3; 4; 5; 6; 7) have been iments (13; 14; 15), they cannot predict catch bonds
6 devoted to study the surprising kinetic and mechanical because force in these models only lowers height of the
0 behaviors of the bonds between L- and P-selectin and energy barrier while shortening lifetimes of the bonds.
/ P-selectin glycoprotein ligand 1 (PSGL-1) at the sin- An exception is the Hookean spring model proposed
o
i gle molecule level: the lifetimes of these bonds first in- by Dembo many years ago (13; 14), in which a catch
b
crease with initial application of small force, which are bond was raised in mathematics. Compared to the ex-
-
q termed“catch”bonds,andsubsequentlydecrease,which ponential decay of the lifetimes of slip-bonds with re-
: aretermed“slip”bondswhenthe forceincreasesbeyond spect to force in experiments (12), the model claimed
v
some thresholds. The most important biological mean- thatthe lifetimes decreaseexponentiallywith the square
i
X ing of this discovery is that the catch-slip transitions of of force (13). In addition, the Hookean spring model
r the PSGL-1 L- and P-selectin bonds may provide a cannot account for the catch-slip bond transitions in
a − −
direct experimental evidence at the single-molecule level self-consistent term. Prompted by the intriguing experi-
to account for the shear threshold effect (8; 9), in which mental observations, three chemical kinetic models have
the number of rolling leukocytes first increases and then been developed. Evans et al. presented a two path-
decreases while monotonically increasing shear stress. ways,twoboundstatesmodelwithrapidequilibriumas-
On the theoretical side, it is a challenge to give a rea- sumption between the two states. They suggested that
sonable physicaltheory or model to explain the counter- the catch-slip bond transitions take place due to applied
intuitivebondtransitions. Bell(10)firstlysuggestedthat force switching the pathways from the one with slower
the force induced dissociation rate of adhesive receptor- dissociation rate to the fast one (4). Although this in-
ligand complex could be described by, sightfulviewpointwelldescribedforcejump-rampexper-
iments, there are two apparent flaws in physics. First
k (f)=k0 exp[fξ‡/k T], (1)
off off B if the forced dissociation experiments were performed
at very low temperatures or higher solvent viscosities,
force would be independent of the bond dissociations
sincetheforceinthetwopathwaysandtwoboundstates
∗Emailaddress:[email protected]
2
model only acts on the inner bound states, while these the small ligand binding to heme proteins (20): there
states would be “frozen” under this circumstances. The is a energy surface for dissociation which dependents on
other is that force does not accelerate the dissociation boththe reactioncoordinatefor the dissociationandthe
processes further when the force is sufficiently large for conformational coordinate x of the complex, while the
the fast dissociation rate is a constant. The next model later is perpendicular to the former; for each confor-
givenbyBarsegovandThirumalai(16)withsamekinetic mation x there is a different dissociation rate constant
scheme seems to improve the two flaws in which the dis- which obeys the Bell rate model, while the distribution
sociation rates of the two pathways were allowed to be of x could be modulated by the force component along
force-dependent with the Bell formulas. Unfortunately, x-direction; higher temperature or larger diffusivity (low
so many independent reaction constants with arbitrary viscosities)allowsxvariationwithinthecomplextotake
dependencies on the force parameters (total seven pa- place, which results in a variation of the energy barrier
rameters) and the final dissociation rate depending on of the bond with time.
them in a complicated way make the physical explana- Therearetwotypesofexperimentalsetupstomeasure
tionsandthedeterminationoftheparametersdifficultto forceddissociationofreceptor-ligandcomplexes. Firstwe
track. Veryrecently,acompetitivetwopathwaysandone consider constant force mode (3; 5). A diffusion equa-
bound state model was proposed by Thomas et al. (17). tion in the presence of a coordinate dependent reaction
This model is distinct from the others because there is a is given by (20)
catch pathway therein, which was thought to arise from
∂p(x,t) ∂2p ∂ ∂V
a backward unbinding pathway. But the model is not =D +Dβ p f⊥ k (x,f )p, (2)
intuitively obvious just like the authors pointed out. ∂t ∂x2 ∂x ∂x − off k
(cid:18) (cid:19)
As one type of noncovalent bonds, interactions of ad- where p(x,t) is probability density for finding a value
hesive receptors and their ligands are weaker. Moreover, x at time t, and D is the diffusion constant. The mo-
the interfaces between them have been reported to be tion is under influence of a force modulating potential
broad and shallow, such as the crystal structure of the V (x) = V(x) f x, where V(x) is intrinsic potential
PSGL-1 P-selectin bond revealed (18). Therefore it is inf⊥the absenice o−f an⊥y force, andi a coordinate-dependent
−
plausible that the energy barriers for the bonds are fluc- Bellrate. InthepresentworkEq.1dependsonxthrough
tuating with time due to either global conformational the intrinsic rate k0 (x), and the distance ξ‡ is assumed
off
changes or local conformational changes at the inter- to be a constant for simplicity. Here f and f are re-
⊥ k
faces. Association/dissociation reactions with fluctuat- spectiveprojectionsofexternalforcef alongthereaction
ingenergybarrierhavebeendeeplystudiedbystatistical and conformational diffusion coordinates:
physicists in terms ofrate processeswith dynamic disor-
der (19)duringthe pasttwodecades. Aprototypeisthe f⊥ = fsinθ, (3)
ligandrebindinginmyoglobinwheretherateconstantde- f = fcosθ 0,
k
≥
pendsonaproteincoordinate(20). Henceitisofinterest
andθistheanglebetweenf andthereactioncoordinate.
to determine whether the fluctuation of energy barrier
We are not ready to study general potentials here. In-
responses to catch-slip bond transitions. Such studies
stead, we focus on specific V(x)s, which make V (x) to
should be meaningful since in the Bell’s initial work and i f⊥
be
the other models developed later, the intrinsic rate con-
stants k0 were deterministic and time-independent. It f⊥
off V (x)=V x η +W(f ), (4)
is possible to derive unexpected results from the relax- f⊥ − − κ ⊥
(cid:18) (cid:19)
ationofthis restriction. Stimulated by the two consider-
where η and κ are two constants with Length and Force
ations, in the present work we propose that the intrinsic
dimensions. For example for a harmonic potential
dissociationrateintheBellmodeliscontrolledbyacon-
formationalcoordinateofreceptor-ligandcomplex,while V(x)=V +k (x x )2/2 (5)
i 0 x 0
−
the coordinate is fluctuating as a Brownian motion in a
with a spring constant k in which we are interested, it
bound harmonic potential; applied force not only lowers x
gives
theheightofthe energybarrierasdescribedinEq.1but
alsomodulatesthedistributionoftheconformationalco- 2
f k f
⊥ x ⊥
ordinate. Inadditiontowellpredictingtheexperimental V x x0 = x x0 (6)
− − k 2 − − k
data,ourtheorymayalsoprovideanewphysicalmecha- (cid:18) x(cid:19) (cid:18) x(cid:19)
nismforthedissociationratessuggestedbyBell(10)and and
Dembo (13) early. f2
W(f )=V f x ⊥ . (7)
⊥ 0 ⊥ 0
− − 2k
x
Defining a new coordinate variabley =x η f /κ, we
I. THEORY AND METHODS − − ⊥
can rewrite Eq. 2 with the specific potentials into
The physical picture of our theory for the forced dis- ∂ρ(y,t) ∂2ρ ∂ ∂V(y)
= D +Dβ ρ k (y)ρ (8)
sociation of receptor-ligand bonds is very similar with ∂t ∂y2 ∂y ∂y − f
(cid:18) (cid:19)
3
where k (y) = k (y+η+f /κ,f ). Compared to the Considering that the system is in equilibrium at the ini-
f off ⊥ k
original work by Agmon and Hopfield (20), our problem tial time, i.e., no reactions at the beginning, the first
fortheconstantforcecaseisalmostsameexceptthereac- eigenvalueλ(0) mustvanish. Onthe otherhand,because
0
tionratenowisafunctionoftheforce. Hence,allresults
obtained previously could be inherited with minor mod- φ00(y)∝exp(−V(y)/2kBT), (16)
ifications. Considering the requirement of extension of
and the square of φ0 is just the equilibrium Boltzmann
Eq. 2 to dynamic force in the following, we present the 0
distributionp (y)withthe potentialV(y),we rewritten
essential definitions and calculations. eq
the first correction of λ (f) as
Substituting 0
V λ(1)(f)= p (y)k (y)dy, (17)
ρ(y,t)=N exp φ(y,t) (9) 0 eq f
0
−k T Z
(cid:18) B (cid:19) p (y) exp[ V(y)/2k T].
eq B
∝ −
into Eq. 8, one can convert the diffusion-reaction equa-
Substituting the above formulaes into Eq. 9, the proba-
tion into Schro¨dinger-like presentation (21).
bility density function then is approximated to
∂φ ∂2φ V
∂t =D∂y2 −Uf(y)φ=−Hf(φ), (10) ρ(y,t)≈N0exp −2k T exp[−λ0(f)t]φ0(f) (18)
(cid:18) B (cid:19)
where N0 is the normalization constant of the density The quantity measured in the constant force experi-
function at t=0, and the “effective” potential ments is the mean lifetime of the bond τ ,
h i
U (y) = U(y)+k (y) (11) ∞ dQ ∞
f f τ = t dt= Q(t)dt, (19)
D 1 ∂V 2 ∂2V h i −Z0 dt Z0
= +k (y).
2kBT "2kBT (cid:18)∂y(cid:19) − ∂y2# f where the survival probability Q(t) related to the prob-
ability density function is given by
WedefineU(y)foritisindependentoftheforcef. Eq.10
can be solved by eigenvalue technique (20). At larger D Q(t) = p(x,t)dx= ρ(y,t)dy
in which we are interested here, only the smallest eigen- Z Z
value λ0(f) mainly contributes to the eigenvalue expan- ≈ exp −t λ(01)(f)+λ(02)(f) . (20)
sion which is obtained by perturbation approach(22): if
h (cid:16) (cid:17)i
the eigenfunctions and eigenvalues of the “unperturbed” In addition to the constant force mode, force could
Schro¨dinger operator be time-dependent, e.g., forceincreasingwitha constant
loading rate in biomembrane force probe (BFP) experi-
∂2 ment(4). Inprinciplethescenariowouldbemorecompli-
= +U(y) (12)
H −∂y2 catedthan that for the constantforce mode. We assume
that the force is loaded slowly compared to diffusion-
in the absence of kf(y) have been known, reactionprocess. Wethenmakeuseanadiabaticapprox-
imation analogous to what is done in quantum mechan-
Hφ0n =−λ0nφ0n, (13) ics. Thecorrectionofthisassumptionwouldbetestedby
theagreementbetweentheoreticalcalculationandexper-
and k is adequately small, the first eigenfunction φ (f)
f 0 imental data. We still use Eq. 2 to describe bond disso-
andeigenvalueλ (f)oftheoperator thenarerespec-
0 f ciations with the dynamic force, therefore we obtain the
H
tively given by
almostsameEqs.4-11exceptthattheforcethereinisre-
placedbyatime-dependentfunctionf . Weimmediately
φ0(f) = φ(00)+φ(01)(f)+··· (14) have (22) t
φ0(y)k (y)φ0 (y)dy
= φ0+ 0 f m φ0 + t
0 mX6=0R λ00−λ0m m ··· φ(y,t)≈exp(cid:20)−Z0 (λ0(ft′)+B(t′))dt′(cid:21)φ0(ft), (21)
and where the “Berry phase”
λ (f) = λ(0)+λ(1)(f)+λ(2)(f)+ (15) ∂
0 0 0 0 ··· B(t)= φ0(ft) φ0(ft)dy, (22)
∂t
= λ0+ φ0(y)k (y)φ0(y)dy+ Z
0 0 f 0
and φ (f ) is the first eigenfunction of the time-
Z 0 t
φ0(y)k (y)φm(y)dy 2 dependent Scho¨dinger operator
0 f 0 + .
λ0 λ0 ···
mX6=0(cid:0)R 0− m (cid:1) Hft = H+kft(y). (23)
4
Because the eigenvalues and eigenfunctions of the above wherethe heightofthe energybarrieralongthe reaction
operatorcannotbesolvedanalyticallyforgeneralk ,we coordinate ∆G‡(x) is a function of the conformational
ft
also apply the perturbation approach. Hence, we obtain coordinate x. According to the form of barrier, we first
φ (f ) andλ (f )by replacingk inEqs.14and 15with analyze two simple and meaningful cases.
0 t 0 t f
k . The Berry phase then is approximated to Bell-like forced dissociations. The simplest function
ft
of the energy barrier might be linear with respect to x,
2
1
B(ft) ≈ m6=0(cid:18)λ0m(cid:19) Z φ00(y)kft(y)φ0m(y)dy× ∆G‡(x)=∆G‡0+kg(x−x0), (32)
X
φ0(y)dkftφ0 (y)dy (24) where ∆G‡0 is the height at position x0, and the slope
0 dt m kg 0 (its dimension Force) for the perturbation re-
Z ≥
quirement in solving Eq. 10. According to Eqs. 15 and
Finally, the survival probability for the dynamic force is
24, we easily get
given by
βk2
Q(t)≈exp − t λ(01)(ft′)+λ(02)(ft′)+B(ft′) dt′ λ(01)(f) = k0exp"−β∆G‡0+ 2kxg#×
(cid:20) Z0 (cid:16) (cid:17) (cid:21)
(25) exp β ξ‡f kgf ,
k− k ⊥
Differentfromtheconstantforcemode,dataofthedy- (cid:20) (cid:18) x (cid:19)(cid:21)
namicforceexperimentsistypicallypresentedintermsof λ(2)(f) = −k02 exp 2β∆G‡+ βkg2 (33)
theforcehistogram,whichcorrespondstotheprobability 0 βDkx "− 0 kx #×
density of the dissociation forces p(f)
k ∞ 1 βk2 n
exp 2β ξ‡f gf g ,
p(f)= dQ df (26) (cid:20) (cid:18) k− kx ⊥(cid:19)(cid:21)n=1nn! kx !
− dt dt X
(cid:30)
and
Particularly, when the force is a linear function of time
f = f + rt, where r is the loading rate, and zero or d k
0 B(f ) = ξ‡f gf
nonzero of f0 respectively corresponds to the steady- t dt(cid:18) tk− kx t⊥(cid:19)×
or jump-ramp force mode in the dynamic force experi-
k2 βk2
ment (4), we have 0 exp 2β∆G‡ + g (34)
βD2kx2 "− 0 kx #×
1
P(f,f0)≈ r λ(01)(f)+λ(02)(f)+B(f) × (27) exp 2β ξ‡f kgf ∞ 1 βkg2 n,
exp 1 f hλ(1)(f′)+λ(2)(f′)+B(fi′) df′ . (cid:20) (cid:18) tk− kx t⊥(cid:19)(cid:21)nX=1n2n! kx !
"−r Zf0 (cid:16) 0 0 (cid:17) # where β = 1/kBT. For large D or kx (or very small T),
the secondcorrectnessandthe Berryphase tendto zero.
Under these limitations the first eigenvalue of Eq. 2 is
II. RESULTS
approximated to be
We consider a bounded diffusion in the harmonic po-
βk2
tential Eq. 5. Then reduces to a harmonic oscillator λ (f) k exp β∆G‡ + g exp[βd‡f]. (35)
operator with H 0 ≈ 0 "− 0 2kx#
Dk k y2 Here we define a new distance
x x
U(y)= 1 . (28)
2k T 2k T −
B (cid:18) B (cid:19) d‡ =ξ‡cosθ ζsinθ, (36)
−
Its eigenvalues and eigenfunctions are
where ζ = k /k whose dimension is Distance. We see
g x
λ0 = nDk /k T (29) that the presence of the complex conformational coordi-
n x B
nate could modify the originalBell model in novel ways:
and (i) d‡ > 0, Eq. 35 is indistinguishable from the origin
φ0(z) = 2−n/2π−1/4(n!)−1/2e−z2/2H (z), (30) Bellmodel,althoughtheprojectiondistanceξ‡cosθfrom
n n the bound state to the energy barrier may be increased
respectively, where z =(k /2k T)1/2y and H (z) is the or decreased in terms of the orientation of the applied
x B n
Hermite polynormials (22). Given that the intrinsic dis- force. In particular, if the force is antiparallel to x, i.e.,
sinθ = 1,wegetaBell-likerateexpressionwitha“dis-
sociation rate satisfies the Arrenhenius form
tance” −ζ; (ii) d‡ = 0, the force does not affect disso-
k0 (x)=k exp ∆G‡(x)/k T , (31) ciations of the bonds, which have been named “ideal”
off 0 − B
(cid:2) (cid:3)
5
bonds by Dembo (13; 14); (iii) d‡ < 0, the force slows ∆G
down dissociations of the bonds. It is “catch” bonds in
whichweareinterested. Incontrasttothecatchbehavior
suggested by Dembo (13), the rate decays exponentially ∆Gb(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
with respect to the force instead of the square of the (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
force. Given the linear function Eq. 32 and f⊥ > 0, in- (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
creasingoftheforceonlystabilizesthebondsbydragging ∆Gs (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
thesystemtothehigherenergybarriers(catch),whereas (cid:0)(cid:1)
theotherforcecomponentfk destabilizesthecomplexby (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)P(cid:0)(cid:0)(cid:1)(cid:1)eq(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) f
lowering the energy barriers(slip). Therefore the sign of (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
the distance d‡ in fact reflects a competition of the two ∆Gc (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
contrast effects of the same force. (cid:0)(cid:1)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
Dembo-like forced dissociations. Another function (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)
(cid:0)(cid:1)
of the energy barrier is a harmonic with a spring con- (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
stant k x x x
g 0 b
∆G‡(x)=∆G‡ +k (x x )2/2, (37)
1 g − 1 FIG.1 Schematicdiagramoftheheightfunctionoftheenergy
where ∆G‡ is the barrier height at position x . Because barrier with respect to the coordinate x. ∆G‡s and ∆G‡c are
1 1 thevalueofthelinearfunctionsinEq.39atpositionx0,while
foranyformofthebarrierheight,thedependenceofλ(02) ∆G‡b istheintersection ofthefunctionsatxb. Theboldsolid
andB(ft)onDisthesamefromEq.29,weonlyconsider lineistheminimummodelwhichisusedtofittheexperiment.
the large D limitation in the following. Hence we have Theshadedarearepresentstheequilibriumdistributionofthe
conformationalcoordinateunderthepotentialV(y)(Eq.17).
1
k 2
λ (f) k x exp β∆G‡ +βξ‡f
0 ≈ 0 k +k − 1 k ×
(cid:18) x g(cid:19)
h i
βk (f k (x x ))2
exp g ⊥− x 1− 0 (38) Fig.1 showsthe characteristicsofthe function. We then
(cid:20)− 2kx(kx+kg) (cid:21) have
Given sinθ > 0 and x x > 0, we find that there is a
interesting transition 1fr−om0slip to catch bond when the λ (f) k0c exp βkc2 exp βd‡f
0 ≈ 2 2k − c
force increase over a threshold kx(x1 x0)/sinθ; other- (cid:20) x(cid:21)
− (cid:2) (cid:3)
wiseonlycatchbondpresents. Wenotethatthelatteris k βk β
c x
erfc ∆+ +f sinθ
very similar to the result proposed by Dembo (13) even
their physical origins are completely different: both of × "−(cid:18) kx(cid:19)r 2 r2kx #
themexponentiallydependentonthesquareoftheforce. + k0s exp βks2 exp βd‡f (41)
Comparison with the experiments. In the constant 2 2k s
(cid:20) x(cid:21)
forceruptureexperimentofthePSGL-1 P-selectincom- (cid:2) (cid:3)
plex,the dissociationrateasthe inverse−meanlifetime of erfc ∆+ ks βkx f β sinθ
the complex firstdecreasedandthen increasedwhen the × "(cid:18) kx(cid:19)r 2 − r2kx #
applied force increasedbeyonda force threshold(3). We
where∆=x x ,andthecomplementaryerrorfunction
now can easily understand this counterintuitive transi- b 0
−
tliikoendaicscsoorcdiaitnigontoratthee:pthreevdioisussodciiasctiuosnsieoffneacbtoouftfthreegBaeinlls- erfc(x)= 2 ∞e−x2dx. (42)
k √π
its dominance when the force is beyond the threshold. Zx
Although in principle we can construct various barrier Before fixing numerical values of the parameters in
height functions which result into catch-slip transitions, Eq. 41, we first simply analyze the main properties of
the mostsimplest formmay be a compositionof twolin- λ (f) given ∆ 0: (i) in the absence of force, due to
0
ear functions erfc( )=2 a≫nd erfc(+ )=0, we have
−∞ ∞
∆G‡(x)=∆G‡+k (x x ), x x
∆G‡(x)=(∆G‡sc(x)=∆G‡bb+kcs(x−−xbb), x≤>xbb (39) λ0 ≈k0cexp(βkc2/2kx), (43)
which is the same with that obtained by Agmon and
wherewerequirethatthedistancesdefinedinEq.36with
Hopfield (20); (ii) if force is nonzero and smaller,
k and k are respectively minus and positive. For con-
c s
venience, their absolute values are correspondingly de- λ kcexp(βk2/2k )exp( βd‡f), (44)
noted by d‡ and d‡. Define two “intrinsic” dissociation 0 ≈ 0 c x − c
c s
constants which means that the bond is catch; and finally (iii),
when the force is sufficiently large, Eq. 41 reduces to
kc =k exp[ β∆G‡(x )],
0 0 − c 0
ks =k exp[ β∆G‡(x )]. (40) λ ksexp(βk2/2k )exp(βd‡f). (45)
0 0 − s 0 0 ≈ 0 s x s
6
1
TABLE I Comparison of the parameters of the present the-
ory, and the two-pathway and one energy-well model pre-
sented by Thomas et al. (17) on the constant force (cf) and 0.8 Theory
sPSGL−1
the dynamic force (df) modes. The parameters for the slip
PSGL−1
behavior of the P-selectin are also listed as a reference (23). )
c
se 0.6
(
e
dc nm ds nm k0c sec−1 k0s sec−1 etim 0.4
Experiment 0.14 0.2 Lif
Dynamicdisorder by us 1.2 0.22 23.2 1.68
Two-pathway one-well (cf) 2.2 0.5 120 0.25 0.2
Two-pathway one-well (df) 0.4 0.2 20 0.34
0
0 10 20 30 40 50
Force (pN)
It is the ordinary slip bond.
There are totaleightindependent parameterspresent-
FIG.2 Themeanlifetimeasafunctionofforceforthebonds
ing in Eq. 41: θ, k , k , k , ∆, ξ‡, k , and ∆G‡(x ).
x c s 0 c 0 of dimeric P-selectin with monomeric sPSGL-1 (square sym-
It is not necessary to determine all of them, which is
bols) (3) and the rescaled dimeric PSGL-1 (circle symbols)
also impossible only through fitting to the experimental
from Ref. (17). The two dash curves are respectively cal-
data (3). For example, the latter two parameters are culated by the two addition terms in Eq. 41 with the same
lumped into kc, while and ξ‡ always presents with cosθ parameters.
0
together. What we reallyconcernwith is the coefficients
of the force and the factors before the error functions
in Eq. 41. They can be obtained by least square fit.
Guided by the properties Eqs 43 and 44, the fitting pro- selectin to the double bonds. A natural assumption is
cess in fact is simple. Even so, we are still able to fix that the two bonds share the same force and fail ran-
all parametersfromthe fitting results if we study a min- domly. The same assumption has been used in previous
imum model in which the slop k is zero and θ = π/6. works (4; 17). Hence, the probability density of the dis-
s
Here the particular value of the angle is actually of no sociation force for the double bond PSGL-1 P-selectin
−
particular significance and it is only as a reference. We complex is related to the single case by
immediately have: k 0.60 pN, k 0.21 pN nm−1,
ξ‡ 0.25 nm, ∆ 33cn≈m; the otherxin≈teresting parame- Pd(f,f0)=P(f/2,f0/2)2. (47)
≈ ≈
ters see Tab. 1, where the values are independent of the
Fig. 3 presents the final result. We see that the theoret-
angle. Substituting these values into Eq. 41 and accord-
ical prediction agrees to the data very well. Hence we
ingtoEq.19,wecalculatethemeanlifetimeofthePSGL-
conclude that the adiabatic approximation proposed at
1 P-selectin complex with respect to different constant
− the beginning is reasonable. The previous works (4; 17)
force in Fig. 2: the agreement between theory and the
have claimed that they could not fit the experimental
experimental data is quite good.
data from the constant force experiment using atomic
More challenging experiments to our theory are the
force microscopy (AFM) and the force jump-ramp ex-
force steady- and jump-ramp modes (4). Under large D
periment using BFP with the same parameters, e.g., see
limitation, Eq. 27 reduces to
Tab. 1. The authors simply contributed it to the dif-
ferent equipment and biological constructs though the
λ (f) 1 f
P(f,f ) 0 exp λ (f′)df′ . (46) experiments studiedthe samecomplexes (3; 4). Ourcal-
0 0
≈ r "−r Zf0 # culations however show that the mechanical parameters
defined by us have almost the same values. In addition,
We see that the mean lifetime can be extracted from
the tendencies of our density functions for the first two
the above equation by setting f = f , i.e., τ =
0 panelsofthesecondarrayinFig.3areclosertothedata
h i
1/rP(f ,f ). We calculate the dissociation force distri-
0 0 thanthat predictedby the two-pathwaysmodels (4; 17).
butions of the steady- and jump-ramp modes at three
The density functions and the force histograms in the
loadingratestocomparewiththe BFPexperimentsper-
experiments reach the maximum and minimum at two
formed by Evans et al. (4). Here we are not ready to
distinct forces, which are named f and f in the
min max
fit the experiments afresh; instead we directly apply the
following,respectively. This observationcould be under-
parameters obtained from the constant force mode to
stood by setting the derivative of Eq. 27 with respect to
current case. Because the BFP experimental data is for
f equal to zero,
dimericligandPSGL-1,whereasourparametersarefrom
monomeric ligand sPSGL-1. Therefore it is necessary dλ
r 0(f)=λ2(f), (48)
to map our predictions for the single bond sPSGL-1 P- df 0
−
7
20 employing the Taylor’s expanding approach we have
steady ramp steady ramp steady ramp
20 pN/sec 200 210 pN/sec 200 1400 pN/sec
1d2λ
λ (f)=λ (f )+ 0(f )(δf)2+o[(δf)3], (50)
10 0 0 c 2 df2 c
100 100
where δf =f f . Substituting it into Eq. 48, we get
c
−
0 0 0
0 100 200 0 100 200 0 100 200 d2λ
f f +λ2(f ) r 0(f ) r−1. (51)
150 min ≈ c 0 c df2 c ∝
jump 28 pN jump 30 pN jump 35 pN (cid:30)
ramp 35 pN/sec 200 ramp 210 pN/sec 200 ramp 1400 pN/sec
100 It means that fmin tends to fc very fast. Different from
f , the loading rate dependence of f is an intrinsic
max min
100 100 property of the catch-slip bond; a unique requirement
50
is that the dissociation rate λ (f) has a minimum at
0
the transition force f . Therefore f s observed in ex-
0 0 0 c min
0 100 200 0 100 200 0 100 200 periment performed by Evans et al. (4) are almost the
Force (pN) Force (pN) Force (pN)
catch-slip transition force observed in the constant force
ruptureexperimentperformedbyMarshallet al.(3). In-
FIG. 3 The probability density of the dissociation forces
Pd(f,f0) under the different loading rates predicted by our deed, the force values of the minimum force histograms
theory (solid curves) for the PSGL-1−P-selectin complex. fortheformerareabout26pN,whilethetransitionforce
Thesymbolsarefromtheforcesteady-andjump-rampexper- for the latter (dimeric PSGL-1 P-selectin)is also about
−
imentaldata(4). Theapparentdeviationsbetweenthetheory 26pN. We know thatthe dissociationforcesdistribution
andthedatainthelastcolumnmaybefromtheinvalidation of a simple slip bond only has a maximum at a certain
of the assumption of two equivalent bonds at higher loading force value that depends on loading rate (25). There-
rates. for the catch-slip bond can easily be distinguished from
the slip case by the presence of a minimum on the den-
sity function of the dissociation forces at a nonvanished
force. Because the above analysis is independent of the
We immediately see that the values of fmin and fmax initial force f0, in order to track the catch behaviors in
must be larger than the catch-slip transition force fc theforcejump-rampexperiments,f0 shouldbechosento
for the left term in Eq. 48 is negative as the bond is be smaller than f .
c
catch. Indeed the experimental observations show that
the force values at the minimum histograms are around
a certain values even the loading rates change 10-fold. III. CONCLUSIONS AND DISCUSSION
(seeFigs.2and4inRef.(4)). Theaboveequationcould
have no solutions when the loading rate is smaller than Compared to the chemical kinetic schemes, our the-
a critical rate rc, which can be obtained by simultane- ory should be more attractive on the following aspects.
ouslysolvingEq.48andits firstderivative. We estimate First of all, we suggest that the counterintuitive catch-
rc 6pN/susingthecurrentparameters,whiletheforce slip transition is a typical example of the rate processes
≈
fmax = fmin 13 pN (about 26 pN in the double bond with dynamic disorder. Because this concept has been
≈
cases). Ifr rc,thenthedensityfunctionismonotonous broadly and deeply studied from theory and experiment
≤
and decreasing function. Therefore, the most probable during the past two decades, extensive experience and
force at the bond dissociation is zero. The most inter- knowledge could be used for reference. For example, we
estingcharacteristicsofEq.48arethedependenceofthe suggestthatanewreceptor-ligandforceddissociationex-
maximum and minimum forces on the loading rate. In periment could be performed over a large range of tem-
particular the latter is an important index in dynamic peratures and solvent viscosities. According to Eq. 2, if
force spectroscopy (DFS) theory since it corresponds to the viscosity is so higher that D 0, we could predict
themostpossibledissociationforce(24). Whentheload- →
ing rate is sufficiently large,and correspondinglyfmax is p(x,t) p(x,0)exp[ tkoff(x,fk)]. (52)
≈ −
larger,the approximationof Eq. 41 at large force Eq. 44
implies that Weknowthatsuchadissociationreactionisatypicalex-
ample of the rate processes with static disorder (19). In
1 βrd‡ addition that the survival probability of the bond con-
fmax ≈ βd lnksexp[βks2/2k ] ∝lnr. (49) verts into multiple exponential decay at a single force
s 0 s x from the single exponential decay at the large D limita-
tion (see Eq. 20), the mean lifetime is
The experimental measurement supported this predic-
tion; see Fig. 3A in Ref. (4). On the other hand, due to
τ p(x,0)k−1(x,f ), (53)
that fmin is very close to fc at the larger loading rate, h i≈ off k
Z
8
which means that the catch-slip bond changes into slip [4] Evans, E., Leung, A., Heinrich, V., & Zhu, C. (2004)
bond only. Then our theory gives a intuitively obvious Proc. Natl. Acad. Sci. USA 101, 11281-11286.
physical explanation of catch bonds in an apparent ex- [5] Sarangapani,K.K.,Yago,T.,Klopocki,A.G.,Lawrence,
pression(Eq.35): they couldarisefromacompetitionof M.B.,Fieger,C.B.,Rosen,S.D.,McEver,R.P.&Zhu,
C. (2003) J. Biol. Chem. 279, 2291-2298.
the two components of applied external force along the
[6] Yago, T., Wu, J. H., Wey, C. D., Klopocki, A. G., Zhu,
dissociation reaction coordinate and the molecular con-
C., & McEver, R.P. (2004) J. Cell. Biol. 166, 913-923.
formational coordinate; the former accelerates the dis-
[7] Marshall,B.T.,Sarangapani,K.K.,Lou,J.Z.,McEver,
sociation by lowering the height of the energy barrier, R. P., & Zhu,C. (2005) Biophys. J. 88, 1458-1466.
while the latter stabilizes the complex by dragging the
[8] Finger, E. B., Puri, K. D., Alon, R. Lawrence, M. B.,
system to the higher barrier height. Finally, the time- von Andrian, U. H., & Springer, T. A. (1996) Nature
dependence of the forced dissociation rates could be in- 279, 266-269.
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complex or localconformationalchanges at the interface K. (1997) J. Cell. Biol.136, 717-727.
between the receptor and ligand; no separated bound [10] Bell, G. I.(1978) Science 200, 618-627.
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Therefore it is possible that one cannot find new sta- ture 374, 539-542.
[12] Chen,S.&Springer,T.A.(2003) Proc. Natl. Acad. Sci.
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USA 98, 950-955.
molecular dynamics (MD) simulations.
[13] Dembo, M., Tourney,D. C., Saxman, K.& Hammer, D.
Eventherearemanyadvantagesinthe presenttheory. (1988) Proc. R. Soc. Lond. B 234, 55-83.
We cannot definitely distinguish which theory or model
[14] Dembo, M. (1994) in Lectures on Mathematics in the
is the most reasonable and more close real situations Life Sciences: Some Mathematical Problems in Biology
with existing experimental data. Moreover, except the eds.Goldstein,B.&Wofsy,C.(Am.Mathematical Soc.,
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[17] Pereverzev, Y. V., Prezhdo, O. V., Forero, M.,
creasing height of the energy barrier with respect to the
Sokurenko,E.V.,&Thomas, W.(2005) Biophys. J. 89,
conformational coordinate to the hook structure (26) or
1446-1454.
more affinity bound states (27), however we believe that
[18] Somers, W. S., Tang, J., Shaw, G. D., & Camphausen,
further single-molecule experiments including microma- R. T. (2000) Cell 103, 467-479.
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morecrystalstructuredataanddetailedMDsimulations [20] Agmon, N. & Hopfield, J. J. (1983) J. Chem. Phys. 78,
from the atomic interactions are essential to elucidate 6947-6959.
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