A general expression for bimolecular association rates with orientational constraints

0.19 MB·English

by  Maximilian Schlosshauer

A general expression for bimolecular association rates
with orientational constraints
Maximilian Schlosshauer∗
Department of Physics, University of Washington, Seattle, WA 98195
David Baker†
4 Department of Biochemistry, University of Washington, Seattle, WA 98195
0
0 Wepresentageneralexpressionfortheassociationrateforpartiallydiffusion-controlledreactions
2 betweensphericalmoleculeswithanasymmetricreactivepatchoneachsurface. Reactioncanoccur
n only if the two patches are in contact and properly aligned to within specified angular tolerances.
a This extends and generalizes previous approaches that considered only axially symmetric patches;
J the earlier solutions are shown to be limiting cases of our general expression. Previous numeri-
6 cal results on the rate of protein–protein association with high steric specificity are in very good
2 agreement with the value computed from our analytic expression. Using the new expression, we
investigate theinfluenceof orientational constraints on the rate constant. Wefind that for angular
] constraintsof ∼5o–15o,atypicalrange forexamplein thecaseof protein–protein interactions, the
h reaction rate is about 2 to 3 orders of magnitude higher than expected from a simple geometric
p model.
-
m
Journal reference: J. Phys. Chem. B 106(46), 12079–12083 (2002).
e
h PACSnumbers: 82.20.Pm,82.39.-k,87.15.Rn,87.15.Vv,87.15.-v
c
.
s
c I. INTRODUCTION In addition to this approach, various attempts have
si been made to quantitatively elucidate the influence of
y orientational constraints and rotational diffusion on the
h The association of two macromolecules, in particular association rate constant. Among the earliest studies,
p the formation of protein–protein complexes, is an ubiq- Sˇolc and Stockmayer derived a formal solution [2] of the
[ uitous process in biology. In the simplest case of the
associationrate constant of spherical molecules with ax-
associating species being modeled as uniformly reactive
1 ially symmetric distributions of reactivity and presented
v spheres, the diffusion-controlled association rate is given numerical results [3] for the simplified case of one of
7 by the classic Smoluchowski result [1], kDC = 4πDR, the molecules being uniformly reactive. Schmitz and
2 where D is the relative translational diffusion constant
Schurr [4] investigated both analytically and numeri-
1 and R denotes the sum of the radii of the molecules.
cally the problem of the reaction between mobile ori-
1
Typically, however, successful complex formation hinges
0 entable spheres, carrying single, axially symmetric reac-
ontheproperrelativeorientationofthereactants,which
4 tive patches on their surface, with localized hemispheri-
0 canberepresentedbymoleculescarryingreactivesurface cal sites on a plane. Shoup et al. [5] introduced a gen-
/ patches that have to come into contact with high steric erally applicably approximative treatment that allowed
s
c specificity for the reaction to occur. simplificationofthecomplexformalsolutionsofSˇolcand
si The simple approachof multiplying the Smoluchowski Stockmayer [3] and Schmitz and Schurr [4] to closed an-
y rate constant for uniformly reactive molecules by the alytical expressions; this approximation was also used
h probabilitythatinarandomencounterthetwomolecules by Zhou [6] in deriving an expression for the association
p
are properly oriented (“geometric rate”) yields rate con- rate when each molecule bears an axially symmetric re-
:
v stants that are commonly several orders of magnitude active patch. All these approaches showed that, because
Xi lowerthantheobservedvalues. Someauthorsattributed ofrelative angularreorientationscausedby translational
this puzzling behavior to the presence of long-range at- androtationaldiffusion,thereductioninassociationrate
r
a tractiveinteractionsbetweenthemoleculesthatnotonly broughtaboutbyorientationalconstraintsissignificantly
generallyspeeduptherateofencounterofthemolecules lessthansuggestedbythereductionintheprobabilityfor
but also help “guide” the molecules into configurations a properly oriented encounter.
close to the proper mutual orientation. The previous analytical treatments, however, impose
only (at most) axially symmetric orientational con-
straints, whereas no analytical treatment has been pre-
sentedthusfarforthegeneralcaseofasymmetricreactive
∗Electronicaddress: [email protected] patches(asintheimportantcaseofstericallyhighlyspe-
†Electronic address: [email protected]; To whom corre-
cificprotein–proteininteractions),wherethepreciserela-
spondence should be addressed. Mailing address: Department of
tiveorientationofthebindingpartnershastobespecified
Biochemistry,UniversityofWashington, Box357350, Seattle, WA
98195. Telephone [phone], Fax[phone]. and appropriately constrained.
2
z
The only numerical estimates for the association rate
constant for this general case stem from Brownian
y
Dynamics simulations, as for example performed by
Northrup and Erickson [9], who consider diffusional as- x
2 Θ
φ
sociation of spherical molecules, each bearing a reactive 2
2 x
patch composed of four contact points in a square ar-
rangement on a plane tangential to the surface of the y
z 2
molecules; reaction is then assumed to occur if three of Sphere 2
the fourcontactpointsarecorrectlymatchedandwithin Θ
a specified maximum distance. The rate constants are B
again found to be about 2 orders of magnitude higher z r
1
thanexpectedfromasimple geometricargument,butas z z2
rel y y
the approach is not analytical, the result is not readily 1
generalizable.
Θ
x
In the following, we present a general expression for 1 Θ 1
the partially diffusion-controlled rate constant k for
DC Θ
two spherical molecules with fully asymmetric binding φ A
φ
patches. The theoretical derivation is given in Sec. II. 1
x
Various aspects of our general expression are investi-
gated in Sec. III, where we demonstrate that previous
approachesare,asexpected,limitingcasesofourgeneral x
rel
treatment(Sec.IIIA),discussthedependenceoftherate
Sphere 1
constantonorientationalconstraints(SectionIIIB),and
compare numerical values obtained from our expression
FIG. 1: Absolute and relative coordinate system describing
with the result of a Brownian Dynamics simulation by
the diffusional motion of the two spheres (see text). For the
Northrup and Erickson[9] (Sec. IIIC).
sakeofclarity,allχandsomeoftheφangleshavebeenomit-
ted in thedrawing.
II. THEORY
tem {x ,y ,z }. The z axis coincides with the
rel rel rel rel
A. Model and coordinate system center-to-centervectorr, whereasthe x axislies in the
rel
plane spanned by r and the z axis of the fixed-space
Ourmodelforbimolecularassociation(seeFig.1)con- coordinate system {x,y,z}. The Euler angles ΨA =
sists of two spherical molecules with radii R1 and R2, (φA,θA,χA) and ΨB = (φB,θB,χB) specify the orien-
respectively, whose relative distance and angular orien- tation of the body-fixed coordinate systems {x1,y1,z1}
tation change by translational and rotational diffusion and {x2,y2,z2} with respect to the coordinate system
with diffusion constants D = D1trans+D2trans, D1rot and {xrel,yrel,zrel}.
Drot. The center of sphere 1 coincides with the origin of
2
afixed-spacecoordinatesystem{x,y,z}. Thepositionof
the center of sphere 2 is specified by the center-to-center B. Reaction condition
vector r whose spherical coordinates with respect to the
fixed-space coordinate system are given by (r,θ,φ). To fully specify the position and orientation of two
Each sphere carries a body-fixed coordinate system, rigid bodies, nine variables are required, for instance,
denotedby{x ,y ,z }and{x ,y ,z },respectively,with as introduced through our absolute coordinate system,
1 1 1 2 2 2
theaxesz andz pointingalongr whenthetwospheres (r,θ,φ,Ψ ,Ψ ). However, for the expression of our re-
1 2 1 2
areperfectlyaligned(andhencez andz canbethought action condition, only five variables, describing the dis-
1 2
of pointing at the “center” of the reactive patch). The tancebetweenthe twospheresandtheir relativeorienta-
orientation of these body-fixed coordinate systems with tion, are needed (see Fig. 2). First, the center-to-center
respect to the fixed-space coordinate system {x,y,z} is distance is parametrized by r. The differences in the
parametrized by sets of Euler angles Ψ = (φ ,θ ,χ ) orientation of the two spheres can be fully captured by
1 1 1 1
and Ψ =(φ ,θ ,χ ). The angles φ and θ , i=1,2,are the differences in the Euler angles Ψ = (φ ,θ ,χ )
2 2 2 2 i i A A A A
the usual azimuthaland polar coordinatesof the z axis, and Ψ = (φ ,θ ,χ ), namely, δθ = |θ −θ |, δφ =
i B B B B A B
whereasχ measurestheanglefromthelineofnodes,de- |φ − φ |, and δχ = |χ − χ |. Finally, we need a
i A B A B
finedtobetheintersectionofthexy andthex y planes, measure for the extent to which the reactive patches on
i i
to the y axis. The set (r,θ,φ,Ψ ,Ψ ) comprises the ab- the spheres are aligned with the center-to-center vector
i 1 2
solute coordinates of the system. r, which can be represented by the sum of the polar an-
For a convenient formulation of the reaction condi- gles θ +θ . To facilitate the subsequent calculations,
A B
tion, we additionally introduce a relative coordinatesys- we replace the conditions on |θ −θ | and θ +θ with
A B A B
3
z where
1
∂2 2 ∂ 1 ∂ ∂ 1 ∂2
∇2r = + + sinθ +
∂r2 r∂r r2sinθ∂θ ∂θ r2sin2θ∂φ2
z2 Sphere 2 (cid:18) (cid:19) (3)
Q istheLaplaceoperatoractingonthecenter-to-centervec-
A torr,expressedinthe sphericalcoordinates(r,θ,φ), and
r the δ , s = x,y,z and i = 1,2, denote an infinitesimal
si
rotation of sphere i about its body-fixed s axis. Equa-
i
Q tion (2) can be viewed as composed of three individual
B
diffusional contributions, namely, the diffusional motion
x
Sphere 1 1 df ofthe center ofmass of sphere 2 relativeto sphere 1 and
dc the rotational diffusion of each sphere.
y
2 As in quantum mechanics, we can define angular mo-
x2 y1 mentum operators Jˆsi = −i~∂/∂δsi as generators of in-
finitesimalrotationsofthespheresabouttheirbody-fixed
axes, and can hence rewrite Eq. (2) as
FIG. 2: The axes and angles relevant to the reaction condi- 0=D∇2rc+DrotJ2c+DrotJ2c, (4)
1 1 2 2
tion, Eqs. (1). The angles θA and θB measure how close the
centerofeachreactivepatch(conincidingwiththebody-fixed where J2 = (−i~)−2(Jˆ2 +Jˆ2 +Jˆ2). Using the basic
axes z1 and z2, respectively) is to thecenter-to-centervector relationsi[7] xi yi zi
r. The angles δφ and δχ denote relative torsion angles of the
twobody-fixedcoordinatesystems(x1,y1,z1)and(x2,y2,z2).
dδ =dθ sinχ −dφ sinθ cosχ ,
To facilitate the visualization of these two angles, the origin xi i i i i i
of the x1 and y1 axes (belonging to the coordinate system of dδyi =dθicosχi+dφisinθisinχi, (5)
sphere1) hasbeenshifted suchastocoincidewiththeorigin
dδ =dφ cosθ +dχ ,
ofthecoordinatesystemofsphere2. Ourreaction condition, zi i i i
Eqs. (1), requires near-optimal alignment, i.e., all angles θA, we can express the operator J2 in terms of the Euler
θB, δφ,and δχ must be below given limits. i
angles Ψ =(φ ,θ ,χ ),
i i i i
1 ∂ ∂
independentconstraintsonθA andθB. Ourreactioncon- Ji2 = sinθ ∂θ sinθi∂θ
dition is therefore i i(cid:18) i(cid:19)
1 ∂2 ∂2 ∂2
+ + −2cosθ . (6)
r =R1+R2 ≡R, sin2θi(cid:18)∂φ2i ∂χ2i i∂φiχi(cid:19)
θ ≤θ0 ,
 A,B A,B (1) The advantage of the formulation of the diffusion equa-
δφ=|φA−φB|≤δφ0, tion, Eq. (2), in terms of the operators Ji2 in Eq. (4),
δχ=|χA−χB|≤δχ0. lies in the fact that the properties of the J2 are well-
i
 known, in particular their eigenfunctions, which are
given by the Wigner rotation matrices Dl (φ,θ,χ) =
mn
e−imφdl (θ)e−inχ [8].
C. Derivation of the rate constant expression mn
The generalsolution to Eq. (4) that obeys the bound-
ary condition at r →∞,
To derive an expression for the association rate con-
stant, we determine the concentration c(r,θ,φ,Ψ ,Ψ ) lim c(r,θ,φ,Ψ ,Ψ )=c =const., (7)
1 2 1 2 0
ofspheres2assolutionofthesteady-statetranslational– r→∞
rotational diffusion equation in the absolute coordi- can therefore be written as a series of products of the
nate system {r,θ,φ,Ψ1,Ψ2} introduced in the preceding eigenfunctions of ∇2r, J2, and J2,
1 2
Sec. IIA,
c(r,θ,φ,Ψ ,Ψ )=c + Amm1n1m2n2
∂c ∂2c ∂2c ∂2c 1 2 0 ll1l2
∂t =0 = D∇2rc +D1rot ∂δ2 + ∂δ2 + ∂δ2 lXl1l2mmX1m2nX1n2
(cid:18) x1 y1 z1(cid:19) ×f (r)Ym(θ,φ)Dl1 (Ψ )Dl2 (Ψ ), (8)
CoMmotion ll1l2 l m1n1 1 m2n2 2
rotationofsphere1
+|D{rozt} ∂2c|+ ∂2c + ∂{2zc , }(2) where
2 (cid:18)∂δx22 ∂δy22 ∂δz22(cid:19) Kl+1/2(ξr)
f (r)= (9)
rotationofsphere2 ll1l2 (ξr)1/2
| {z }
4
are the modified Bessel functions of the third kind [10] The absolute coordinate system {x,y,z} can be trans-
(giving the desired behavior f (r) → 0 as r → ∞), formedintotherelativecoordinatesystem{x ,y ,z }
ll1l2 rel rel rel
with ξ ≡[(Drot/D)l (l +1)+(Drot/D)l (l +1)]1/2. by rotations through the three Euler angles (φ−π,θ,0).
1 1 1 2 2 2
For the boundary condition at r = R, the usual, but The corresponding transformations of the rotation ma-
analytically hardly tractable radiation boundary condi- trices appearing in Eq. (13) are then
tion is
DlA (Ψ )= DlA (φ−π,θ,0)DlA (Ψ ),
∂c κ mAnA A m1mA m1nA 1
= F(ΨA,ΨB)c(R,θ,φ,Ψ1,Ψ2), (10) Xm1
∂r D
(cid:12)R DlB (Ψ )= DlB (φ−π,θ,0)DlB (Ψ ).
(cid:12) mBnB B m2mB m2nB 2
whereκqua(cid:12)(cid:12)ntifiestheextentofdiffusioncontrolinthere- Xm2
action,andF(ΨA,ΨB)≡H(θA0−θA)H(θB0 −θB)H(δφ0− The expansion coefficients Amm1n1m2n2 in Eq. (8)
δφ)H(δχ −δχ)representsthereactionconditionEq.(1), ll1l2
0 can be obtained by substituting the expansion for
where H(x) is the step function defined by H(x)=0 for
F(Ψ ,Ψ ), Eq. (13), expressed in absolute coordinates
x<0 and H(x)=1 for x≥0. A B
(r,θ,φ,Ψ ,Ψ ), using the above transformations, into
In our approach, we express the radiation boundary 1 2
Eq. (11), which yields
condition using the constant-flux approximation as in-
troducedbyShoupet al. [5],byrequiringthattheflux is Amm1n1m2n2 = Q (−1)m+m1+m2−n1−n2
a constant over the angularranges in which the reaction ll1l2 f′ (R)
ll1l2
can take place,
× 4π(2l+1) l l1 l2
m−m1 −m2
∂c
∂r =QF(ΨA,ΨB), (11) × p Clm1lA2−n1(cid:0)0l ml1A −ml2A ,(cid:1) (15)
(cid:12)(cid:12)R XmA (cid:0) (cid:1)
and that Eq. (10) is(cid:12)(cid:12)obeyed on the averageover the sur- where l l1 l2 is the Wbigner 3-j symbol. Evaluating
faces of the spheres, that is, mm1 m2
Eq.(12)usingthe expansioncoefficients,Eq.(15),yields
(cid:0) (cid:1)
for the constant Q
dΩ dΨ dΨ F(Ψ ,Ψ )Q
1 2 A B
D f (R)
Z Z Z Q = c a × a − ll1l2 4π(2l+1)
κ 0 o κ 0 f′ (R)
= D dΩ dΨ1 dΨ2F(ΨA,ΨB)c(R,θ,φ,Ψ1,Ψ2), " lXl1l2 ll1l2
Z Z Z (12) × 2l1+12l2+1 +l1 +l1 Cmn l l1 l2 2 (−116)
8π2 8π2 l1l2 0m−m
where we have introduced the abbreviation dΩ ≡ #
sinθdθ dφ. nX=−l1(cid:2)mX=−l1 b (cid:0) (cid:1)(cid:3)
R where we have introduced
To proceed,we expressF(Ψ ,Ψ )in absolute coordi-
A B
R R
nates. First, we expand F(Ψ ,Ψ ) in terms of rotation
A B a = dΩ dΨ dΨ F(Ψ ,Ψ )
0 1 2 A B
matrices,
Z Z Z
= (4π)3δφ δχ (1−cosθ0)(1−cosθ0). (17)
F(Ψ ,Ψ ) = CmAnAmBnB 0 0 A B
A B lAlB a /(4π×8π2×8π2)isthefractionofangularorientational
lXAlBmXAnAmXBnB 0
space over which the reaction can occur. In deriving
× DlA (Ψ )DlB (Ψ ), (13)
mAnA A mBnB B Eqs.(15)and(16),wehavemadeuseoftheidentities[8]
where the expansion coefficients CmAnAmBnB are given
lAlB 2l+1
by Ym∗(θ,φ)= Dl (φ,θ,0),
l 4π m0
r
2l +12l +1
ClmAAlBnAmBnB ×= DlA8Aπ∗2 (ΨB8π)D2lB∗Z d(ΨΨA )ZF(dΨΨB,Ψ ) Z dΨDml1∗1n1(Ψ)Dml22n2(Ψ)= 2l81π+21δl1l2δm1m2δn1n2,
mAnA A mBnB B A B dΨDl1 (Ψ)Dl2 (Ψ)Dl3 (Ψ)
2lA+12lB+1 m1n1 m2n2 m3n3
= Z
8π2 8π2 =8π2 l1 l2 l3 l1 l2 l3 ,
4πsin(m δφ )4πsin(n δχ ) m1 m2 m3 n1 n2 n3
× θA0 mAA 0 nAA 0 mX1m2(cid:0)ml11 ml22 ml33(cid:1)(cid:0)ml11 ml22(cid:0)ml′3′3(cid:1)= 2l3(cid:1)1(cid:0)+1δl3l′3δm(cid:1)3m′3.
× sinθAdθAdlmAAnA(θA) The diffusion-controlled rate constant is given by
Z0
× Z20lθB0+si1n2θlBd+θB1dl−BmA−nA(θB) kDC = (8π112)2RRc220DD Z dΩZ dΨ1 Z dΨ2 ∂∂rc(cid:12)(cid:12)(cid:12)R
≡ A8π2 B8π2 ClmAAlBnA. (14) = (8π2)2 c0 a0Q. (cid:12) (18)
b
5
Since the functions f (r), defined in Eq. (9), obey the δχ =π and θ0 =π), we arriveatthe model introduced
ll1l2 0 B
recursion relation by Sˇolc and Stockmayer [2]. Then, since θ0 = π and
B
π
sinθdθP (cosθ) = 0 if l 6= 0, only the term l = 0
fl′l1l2(r)= rlfll1l2(r)−ξf(l+1)l1l2(r), (19) (Ra0nd hencell = l1) gives a nonzero contribution t2o the
sum in Eq. (21). Then Eq. (21) reduces to
the final expression for the diffusion-limited rate con-
stant, Eq. (18), becomes D
k = 2πDR2(1−cosθ0)2× (1−cosθ0)
DC A κ A
D (cid:20)
k = D(Ra /8π2)2× a (l+1/2)K (ξ∗)
DC 0 "κ 0 −R l+1/2
lK (ξ∗)−ξ∗K (ξ∗)
l+1/2 l+3/2
−R lK (Kξ∗l)+−1/2ξ(∗ξK∗) (ξ∗) XθAl0 2 −1
l+1/2 l+3/2 × sinθ dθ P (cosθ ) , (22)
lXl1l2 A A l1 A
2l +12l +1 (cid:18)Z0 (cid:19) (cid:21)
1 2
× 4π(2l+1)
8π2 8π2 where now ξ∗ = R[(Drot/D)l(l+1)]1/2, which coincides
+l1 +l1 2 −1 with the result of Sˇol1c and Stockmayer [3] and Shoup
× Clm1ln2 0l ml1 −l2m , (20) et al. [5].
with ξ∗ =ξR.nX=−l1(cid:18)mX=−l1 b (cid:0) (cid:1)(cid:19) # θB0As=sumδφin0g b=othδχsp0he=resπt,o boneluynitfhoermtleyrmrealcti=ve,lθ1A0 ==
l = 0 contributes, and hence ξ∗ = 0. Because
2
K (ξ∗)/ξ∗K (ξ∗) → 1 as ξ∗ → 0, Eq. (22) be-
1/2 3/2
III. RESULTS comes, in the fully diffusion-controlled case (κ → ∞),
k = 4πDR, which is just the classic Smoluchowski
DC
A. Limiting cases diffusion-controlledrateconstantfortwouniformlyreac-
tive spheres.
1. Axially symmetric reactive patches
Zhou [6] presented an analyticalexpressionfor the as- B. Numerical evaluation
sociationrateconstantoftwosphericalmoleculesbearing
axiallysymmetricpatches. Inthe notationofourmodel,
In the following, we shall assume the reaction to be
this corresponds to setting δφ =δχ = π, which makes
0 0 fully diffusion-controlled (κ →∞), and take the radii of
Cmn =0inEqs.(14)and(20),unless m=n=0. Using
l1l2 thetwospherestobeidentical,R1 =R2. Insteadofplot-
D0l0(φ,θ,χ) = dl00(φ,θ,χ) = Pl(cosθ), where Pl(cosθ) ting the absolute value of the association rate constant
abretheLegendrepolynomials,theexpressionfortherate k , we introduce the dimensionless relative association
DC
constant, Eq. (20), becomes rate constant k∗ = k /4πDR, which is the ratio of
DC DC
the orientation-constrained rate constant to the Smolu-
k = 4πDR2(1−cosθ0)2(1−cosθ0)2
DC A B chowskirateconstantfortwouniformlyreactivespheres.
D
× 4 (1−cosθ0)(1−cosθ0) The full dependence of the relative association rate
κ A B constant on θ0, θ0, δφ , and δχ is not easy to display
(cid:20) A B 0 0
K (ξ∗) in a single plot. For simplicity, we set all four param-
l+1/2
−R lK (ξ∗)−ξ∗K (ξ∗) eters equal, and in Fig. 3 plot the relative association
lXl1l2 l+1/2 l+3/2 rate constant kD∗C computed from Eq. (20) as a func-
× (2l+1)(2l +1)(2l +1) tion of a single parameter (referred to Φ in the follow-
1 2 0
θA0 2 ing). For comparison, we also show the relative asso-
× sinθ dθ P (cosθ ) ciation rate expected from a purely probabilistic argu-
A A l1 A
(cid:18)Z0 (cid:19) ment (geometric rate), given by the fraction of angular
× θB0 sinθBdθBPlB(cosθB) 2 0l l01 l02 2 −(211) oar0i/e(n4tπat×io8nπal2×spa8cπe2)o=veΦr 0w(1hi−chcotshΦe0r)e2a/c4tπio2n. can occur,
(cid:18)Z0 (cid:19) (cid:0) (cid:1) (cid:21) ItisevidentfromFig.3thatthedifferencebetweenthe
which agrees with the solution presented by Zhou [6]. rateconstantk∗ andthegeometricrategetsmorestrik-
DC
ingastheangularconstraintΦ becomesmorestringent.
0
For instance, in the important case of sterically highly
2. Uniform reactivity specific protein–protein interactions where Φ will typ-
0
ically range between 5o and 15o, the geometric rate is
Ifweassumethatonesphereisuniformly reactiveand about 2 to 3 orders of magnitude too low, as compared
the other has an axially symmetric patch (that is, δφ = with the association rate computed from Eq. (20).
0
6
1 angles θ0, θ0, δφ , and δχ (as defined in our model,
A B 0 0
0.1 see Sec. II) that correspond to this contact-based reac-
tion condition. Clearly, there will be a multiplicity of
0.01
sets of these angles for which the contact-based reaction
0.001
criterion is met. To reduce the search space in a reason-
C 1e-04 able way, we looked for geometric configurations where
∗kD 1e-05 all four angles were equal, θA0 = θB0 = δφ0 = δχ0, and
found that the contact-based reaction condition can be
1e-06
well represented by an angular constraint of θ0 = θ0 =
1e-07 A B
δφ =δχ =6.7o.
kDC fromEq.(20) 0 0
1e-08 geometricrateΦ0(1−cosΦ0)2/4π2 With these angular constraints, numerical evaluation
1e-09 of Eq. (20) with the parameters specified above and
0 20 40 60 80 100 120 140 160 180
Φ0 (degs) κ→∞giveskDC =1.04×105M−1 s−1,whichis invery
good agreement with the value obtained from the Brow-
FIG. 3: Diffusion-controlled (κ → ∞) relative associa- nian dynamics simulation by Northrup and Erickson[9],
tion rate constant kD∗C = kDC/4πDR, with kDC computed k =1×105M−1 s−1.
from Eq. (20), as a function of the angular constraint Φ0 ≡ DC IV. SUMMARY
θA0 = θB0 = δφ0 = δχ0 (solid curve). Also shown is the
rate expected from a simple probabilistic argument, kDC =
Φ0(1−cosΦ0)2/4π2 (geometric rate; dashed curve). We have presented a general expression for the
diffusion-controlled association rate of two molecules
wherereactioncanoccursolelyifspecifiedconstraintson
C. Comparison against Brownian dynamics themutualorientationarefulfilled. Oursolutiongoesfar
simulations beyondprevioustreatmentsinthe abilitytoimposevery
general, asymmetric orientational constraints, as needed
In the Brownian dynamics simulations by Northrup forinstanceinaproperdescriptionofthestericallyhighly
and Erickson [9], protein molecules are modeled as hard specific associationof two proteins.
spheres of R = 18 ˚A diffusing in water (η ≃ 8.9 × Since our expression for the rate constant, Eq. (20),
10−4Ns/m2) at T = 298 K; no forces are assumed to was derived under the assumption of no forces acting
act between the molecules. The translational and rota- between the two molecules, a comparison of measured
tionaldiffusionconstantsarecomputedfromtheStokes– association rates with their theoretical values calculated
Einstein relations Dtrans = k T/6πηR and Drot = from Eq. (20) should reveal the extent to which long-
B
k T/8πηR3, respectively. rangeinteractionscontributetotherateofintermolecular
B
Instead of angular constraints, the model uses a association. Suchaninvestigationwouldbeofparticular
contact-basedreactioncondition. Asetoffourdistinctly interest in the case of the association of proteins with
numbered contact points is mounted on each sphere in small ligands and other proteins.
a 17 ˚A×17 ˚A square arrangement on a plane tangen-
tial to the surface of the sphere. Reaction is assumed
to occur when at least three of the four contact points Acknowledgments
arecorrectlymatchedandwithinamaximumdistanceof
2 ˚A. ThisworkwassupportedbyagrantfromtheNational
We performed numerical simulations to estimate the Institute of Health.
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