Optimal flexibility for conformational transitions in macromolecules
Optimal flexibility for conformational transitions in macromolecules
Richard A. Neher1, Wolfram Möbius1, Erwin Frey1, and Ulrich Gerland1,2
1Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for Nanoscience (CeNS),
LMU München, Theresienstraße 37, 80333 München, Germany
2Institute for Theoretical Physics, University of Cologne, Germany
(Dated: November 20, 2018)
Conformational transitions in macromolecular complexes often involve the reorientation of lever-
like structures. Using a simple theoretical model, we show that the rate of such transitions is
drastically enhanced if the lever is bendable, e.g. at a localized “hinge”. Surprisingly, the transition
is fastest with an intermediate flexibility of the hinge. In this intermediate regime, the transition
rate is also least sensitive to the amount of “cargo” attached to the lever arm, which could be
exploited by molecular motors. To explain this effect, we generalize the Kramers-Langer theory for
multi-dimensional barrier crossing to configuration dependent mobility matrices.
Many biological functions depend on transitions in the
global conformation of macromolecules, and the associ-
ated kinetic rates can be under strong evolutionary pres-
sure. For instance, the directed motion of molecular mo-
tors is based on power strokes [1], protein binding to
DNA can require DNA bending [2] or spontaneous partial
unwrapping of DNA from histones [3, 4], and the func-
tioning of some ribozymes depends on global transitions
in the tertiary structure [5]. These and other examples
display two generic features: (i) A long segment within
the molecule or complex is turned during the transition,
e.g. an RNA stem in a ribozyme, the DNA as it unwraps
from histones or bends upon protein binding, or the lever
arm of a molecular motor relative to the attached head.
(ii) The segment has a certain bending flexibility. Here,
we use a minimal physical model to study the coupled
dynamics of the transition and the bending fluctuations.
Our model, illustrated in Fig. 1, demonstrates explic-
itly how even a small bending flexibility can drastically
accelerate the transition. Furthermore, if the flexibil-
ity arises through a localized “hinge”, e.g. in the protein
structure of some molecular motors [6, 7] or an interior
loop in an RNA stem, we find that the transition rate
is maximal at an intermediate hinge stiffness. Thus, in
situations where rapid transition rates are crucial, molec-
ular evolution could tune a hinge stiffness to the optimal
value. We find that an intermediate stiffness is optimal
also from the perspective of robustness, since it renders
the transition rate least sensitive to changes in the drag
on the lever arm, incurred e.g. by different cargos trans-
ported by a molecular motor.
Our finding of an optimal rate is reminiscent of a phe-
nomenon known as resonant activation [8, 9], where a
transition rate displays a peak as a function of the charac-
teristic timescale of fluctuations in the potential barrier.
However, we will see that the peak in our system has
a different origin: a trade-off between the accelerating
effect of the bending fluctuations and a decreasing av-
erage mobility of the reaction coordinate. The standard
Kramers-Langer theory [10] for multi-dimensional transi-
tion processes is not sufficient to capture this trade-off. A
generalization of the theory to the case of configuration-
dependent mobility matrices turns out to be essential to
understand the peak at intermediate stiffness.
Model.— We model the conformational transition as
a thermally activated change in the attachment angle ϕ
of a macromolecular lever, see Fig. 1. The lever has two
segments connected by a hinge with stiffness ǫ, which ren-
ders the lever preferentially straight, but allows thermal
fluctuations in the bending angle θ. The energy function
V (ϕ, θ) of this ‘Two-Segment Lever’ (TSL) is
V (ϕ, θ)
= ǫ(1− cos θ)−
(aϕ)3
− b(aϕ)
, (1)
where kBT is the thermal energy unit. The hinge, de-
scribed by the first term, serves not only as a sim-
ple model for a protein or RNA hinge, but also as a
zeroth-order approximation to a more continuously dis-
tributed flexibility; see below. The second term is the
potential on the attachment angle ϕ, which produces a
metastable minimum at (ϕ, θ) = (0, 0). The thermally-
assisted escape from this minimum passes through the
transition state at (ϕ, θ) = (b/a, 0) with a barrier height
∆V = b3kBT/6 [20].
a) b)
FIG. 1: Schematic illustration of the ‘Two-Segment Lever’
(TSL) model for conformational transitions. (a) The two
segments of lengths 1 and ρ are connected by a hinge and
attached to the origin. The viscous drag acts on the ends
of the segments as indicated by the beads. (b) Schematic
illustration of the barrier crossing processes. The external
meta-stable potential V (ϕ) is indicated by shading (top; dark
corresponds to high energy) and is also sketched below.
http://arxiv.org/abs/0704.1546v1
In the present context, inertial forces are negligible,
i.e. it is sufficient to consider the stochastic dynamics of
the TSL in the overdamped limit. We localize the fric-
tion forces to the ends of the two segments, as indicated
by the beads in Fig. 1(a). The length of the first segment
defines our length unit and ρ denotes the relative length
of the second segment. Similarly, we choose our time
unit such that the friction coefficient of the first bead is
unity, and denote the coefficient of the second bead by
ξ. To describe the Brownian dynamics of the TSL, we
derive the Fokker-Planck equation for the time-evolution
of the configurational probability density p(ϕ, θ, t). In
general, the derivation of the correct dynamic equations
can be a nontrivial task for stochastic systems with con-
straints [11, 12]. For instance, implementing fixed seg-
ment lengths through the limit of stiff springs, leads to
Fokker-Planck equations with equilibrium distributions
that depend on the way in which the limit is taken [12].
However, for our overdamped system, we can avoid this
problem by imposing the desired equilibrium distribu-
tion, i.e. the Boltzmann distribution p = exp(−V/kBT ),
which together with the well-defined deterministic equa-
tions of motion uniquely determines the Fokker-Planck
equation for the TSL.
The deterministic equations of motion take the form
q̇k = −Mkl ∂V/∂ql with the coordinates (q1, q2) = (ϕ, θ)
and a mobility matrix M. We obtain M with a standard
Lagrange procedure: Given linear friction, M is the in-
verse of the friction matrix, which in turn is the Hessian
matrix of the dissipation function [13]. This yields
1 + ξ sin2 θ
1 ρ+cos θ
ρ+cos θ
ρ+2 cos θ
+ 1+ξ
. (2)
The Fokker-Planck equation then follows from the conti-
nuity equation ∂tp({qi}, t) = −∂kjk({qi}, t) together with
jk({qi}, t) = −Mkl
+ kBT
p({qi}, t) (3)
as the probability flux density. Our analytical analysis
below is based directly on Eqs. (2) and (3), while we
perform all Brownian dynamics simulations with a set of
equivalent stochastic differential equations [14].
Transition rate.— To explore the phenomenology of
the TSL, we performed simulations to determine its av-
erage dwell time τ in the metastable state, for a range of
hinge stiffnesses ǫ. The rate for the conformational tran-
sition is related to the dwell time by k(ǫ) = 1/τ(ǫ). Fig. 2
shows k(ǫ) (circles) for a barrier ∆V =12 kBT , a distance
∆ϕ=0.4 to the transition state, and ξ= ρ=1 (data for
different parameter values behaves qualitatively similar,
as long as the process is reaction-limited, i.e. ∆V is suf-
ficiently large that τ is much longer than the time for the
TSL to freely diffuse over an angle ∆ϕ). We observe a
significant flexibility-induced enhancement of the transi-
tion rate over a broad range of stiffnesses, compared to
0 10 20 30 40 50 60 70
stiffness ε [kT]
Langer rate
gen. Langer rate
0 50 100 150
stiff limit
FIG. 2: Simulation data of the barrier crossing rate normal-
ized by k0 display a prominent peak at finite stiffness (cir-
cles, each obtained from 20000 simulation runs initialized at
the metastable minimum). The conventional Langer theory
fails to describe the non-monotonicity of the rate and over-
estimates the rate at small ǫ. The generalized Langer theory
captures the non-monotonicity of the rate and describes the
simulations data accurately; parameters see main text.
the dynamics in the stiff limit (ǫ→∞), see inset. Note
that the enhancement persists even at relatively large ǫ,
where typical thermal bending fluctuations δϕ ∼
ǫ are
significantly smaller than ∆ϕ. Surprisingly, the acceler-
ation is strongest at an intermediate stiffness (ǫ ≈ 10).
This observation suggests that the stiffness of molecular
hinges could be used, by evolution or in synthetic con-
structs, to tune and optimize reaction rates.
When the friction coefficient ξ of the outer bead is
increased, the rate of the conformational transition de-
creases; see Fig. 3a. This decrease is most dramatic in
the stiff limit (dash-dotted line). In the flexible limit (dia-
monds) the decrease is less pronounced. Notably, the rate
appears least sensitive to the viscous drag on the outer
bead at intermediate ǫ (circles). Indeed, Fig. 3b shows
that the ǫ-dependence of this sensitivity (measured as the
slope of the curves in Fig. 3a at ξ=1) has a pronounced
minimum at ǫ ≈ 20. Hence, intermediate hinge stiffnesses
in the TSL lead to maximal robustness, which is an im-
portant design constraint for many biomolecular mecha-
nisms in the cellular context. For instance, as molecular
motors transport various cargos along one-dimensional
filaments, it may be advantageous to render their speed
insensitive to the cargo size, e.g. to avoid “traffic jams”.
In the remainder of this letter, we seek a theoretical
understanding of the above phenomenology. First, it is
instructive to consider simple bounds on the transition
rate. An upper bound is obtained by completely elimi-
nating the outer bead. The Kramers rate [15] for the re-
maining 1D escape process, k0 = (a
2b/2π) e−∆V/kBT , is
used in Figs. 2 and 3 to normalize the transition rates. At
the optimal stiffness, the transition rate in Fig. 2 comes
within 20% of this upper bound. An obvious lower bound
is the stiff limit: For ǫ→∞, the second segment increases
1 10 100
friction ξ
0 20 40 60 80
stiffness ε [kT]
a) b)
 ε=25
stiff
FIG. 3: The sensitivity of the rate to the friction coefficient
ξ is minimal at intermediate stiffness. (a) Simulation results
at ǫ = 0 and ǫ = 25 as well as the theoretical estimates of the
rate at ǫ = 0 and in the stiff limit. (b) The derivative of ln k
with respect to ln ξ evaluated at ξ = 1, i.e. the slope of the
curves in a), is minimal in an intermediate stiffness range.
the rotational friction by a factor ζ = 1 + (1 + ρ)2ξ, so
that the 1D Kramers rate becomes k∞ = k0/ζ, as shown
by the dash-dotted line in Fig. 2 and Fig. 3a. However,
to understand how the dynamics of the bending fluctua-
tions affects the transition rate, we must consider the full
2D dynamics of the TSL. The multi-dimensional gener-
alization of Kramers theory is Langer’s formula for the
escape rate over a saddle in a potential landscape [10],
kLanger =
det e(w)
| det e(s)|
. (4)
Here, e(w) and e(s) denote the Hessian matrix of the po-
tential energy, ∂2V/∂qk∂ql, evaluated at the well bot-
tom and the saddle point, respectively, whereas λ is the
unique negative eigenvalue of the product of the mobility
matrix M and e(s). Eq. (4) can be made plausible in sim-
ple terms: Given a quasi-equilibrium in the metastable
state, the second factor represents the probability of be-
ing in the transition region, i.e. the region within ∼ kBT
of the saddle. The escape rate is then given by this prob-
ability multiplied by the rate λ at which the system re-
laxes out of the transition state, analogous to Michaelis-
Menten reaction kinetics.
For our potential (1), the determinants in (4) can-
cel. The eigenvalue can be determined analytically (the
dashed line in Fig. 2 shows the resulting kLanger), but for
the present purpose it is more instructive to consider the
expansions for large and small stiffness. In the stiff limit,
the natural small parameter is the stiffness ratio γ/ǫ,
where γ = a2b is the absolute curvature or “stiffness” of
the external potential at the transition state. The ex-
pansion yields kLanger/k∞ = 1+ (ρ
2ξ/ζ) γ/ǫ+O(γ2/ǫ2).
As expected, the rate approaches k∞, but the stiff limit
is attained only when the bending fluctuations ∼
ǫ are
small compared to the width of the barrier ∼ √γ. In the
opposite limit, ǫ≪ γ, the rate is given by kLanger/k0 =
1 + ρ−1
ǫ/γ + O(ǫ2/γ). Since the linear term is
negative, Langer theory predicts that the transition rate
peaks at zero stiffness, with a peak value equal to the
Kramers rate k0 for the lever without the second segment.
a) b)
FIG. 4: The friction opposing rotation of the attachment an-
gle ϕ depends on the bending angle θ, since the outer bead is
moved by different amounts in different configurations. For an
infinitesimal displacement dϕ, the displacement of the outer
bead is sin θ dϕ. The projection of the resulting friction force
onto the direction of motion adds another factor sin θ, yielding
a friction coefficient for ϕ of 1 + ξ sin2 θ.
This prediction is clearly at variance with the simulation
results. It is interesting to note, however, that the slope
of the linear decay is independent of ξ. This is consistent
with our observation that the transition rate is insensi-
tive to ξ in the intermediate stiffness regime. Indeed,
Fig. 2 shows that Langer theory (dashed line) describes
the simulation data (circles) reasonably well for interme-
diate and large hinge stiffness.
To understand the origin of the peak at intermediate
stiffness, it is useful to consider the flexible limit (ǫ = 0).
In this limit, the transition state is degenerate in θ, and
it seems plausible to estimate the transition rate by using
a θ-averaged mobility for the reaction coordinate ϕ,
k(ǫ = 0) ≈ k0
M11(θ) =
1 + ξ
. (5)
This estimate agrees well with the simulation data,
see the dashed line in Fig. 3a, indicating that the
configuration-dependent mobility (2) plays an important
role for the transition rate. In contrast, the conventional
Langer theory assumes the mobility matrix to be con-
stant in the relevant region near the transition state.
Fig. 4 illustrates why the mobility M11 of the coordinate
ϕ is affected by the bending angle θ and gives a graphical
construction for M11.
Generalized Langer theory.— To account for the mo-
bility effect identified above, we must generalize the
Langer theory to configuration-dependent mobility ma-
trices. The special case where the mobility varies only
along the reaction coordinate has already been studied in
[16], however the main effect in our case is due to the vari-
ation in the transverse direction. In the following, we out-
line the derivation of the central result, while all details
will be presented elsewhere. Near the saddle, the mobil-
ity matrix takes the form Mij({qi}) = M (s)ij + 12A
ij q̂lq̂k,
where q̂i are deviations from the saddle and A
ij denotes
the tensor of second derivatives of the mobility matrix
(we assume that the first derivatives of M vanish at the
saddle, which is the case for the TSL). The escape rate
is given by the probability flux out of the metastable
well, divided by the population inside the well. To cal-
culate the flux, we construct a steady state solution to
the Fokker-Planck equation in the vicinity of the saddle,
as described in [15] for the conventional Langer theory.
We use the Ansatz p({qi}) = 12peq({qi}) erfc(u), where
peq({qi}) = Z−1e−V ({qi})/kBT and erfc(u) is the comple-
mentary error function with argument u = Uk q̂k. Insert-
ing the Ansatz into the Fokker-Planck equation yields an
equation for the vector U,
Ui(−Mije(s)jk +Bik)− UiMijUj Uk = 0 , (6)
where Bik = kBT
ni . Bik q̂k is the noise induced
drift, which is absent in the conventional Langer theory.
Ignoring higher order terms, this equation determines U
to be the left eigenvector of −M(s)e(s)+B to the unique
positive eigenvalue λ, and requires U to be normalized
such that UiM
ij Uj = λ. The directions of the left and
right eigenvectors of −M(s)e(s) + B have a physical in-
terpretation: U is perpendicular to the stochastic sepa-
ratrix, while the corresponding right eigenvector points
in the direction of the diffusive flux at the saddle [17].
From p({qi}), the flux density is determined by (3) and
the total flux is obtained by integrating the flux density
over a plane containing the saddle; a convenient choice is
the plane u = 0. Evaluation of the integral is particularly
simple in a coordinate system, where the first coordinate
is parallel toU, and the remaining coordinates are chosen
such that e(s) is diagonal in this subspace, e
ij = µiδij
for i, j > 1. In this coordinate system, the generalized
Langer rate takes the simple form
1 + 1
det e(w)
| det e(s)|
T , (7)
where c = Uie
ij Uj + 1 = B1ie
i1 /M
11 and e
−1 denotes
the inverse matrix of e(s). Eq. (7) contains three correc-
tions to (4), all of which vanish whenM({qi}) is constant:
The most important one is given by
l>1 A
11/µl, which
changes the mobility M11 in the direction of U to an ef-
fective mobility that is averaged over the separatrix with
respect to the Boltzmann distribution. In addition, there
are two corrections incurred by the noise-induced drift:
the factor
1− c and a change due to the fact that λ is
now the eigenvalue to M(s)e(s) −B instead of M(s)e(s).
The solid line in Fig. 2 shows the application of the
generalized Langer formula to the TSL. We observe that
it captures the peak in the transition rate and thus the es-
sential phenomenology of the TSL. Obviously, the evalu-
ation of the Gaussian integral that leads to Eq. (7) is only
meaningful, if the harmonic approximation of the mobil-
ity matrix is reasonable within the relevant saddle point
region. This integral diverges as the saddle point degen-
erates, which explains the behavior for ǫ → 0. At high
ξ, the very anisotropic friction can also render Langer
theory invalid [18, 19].
Conclusion.— We have introduced the “Two-Segment
Lever” as a simple model for a class of conformational
transitions in biomolecules. The model clearly demon-
strates how flexibility can enhance the rate of a confor-
mational transition. This remains true, if the hinge in the
TSL is replaced by a more continuous bendability. In-
terestingly, a discrete hinge has a stiffness regime, where
the rate is large and robust against cargo variation, which
raises the question, whether these effects are exploited by
evolution, for example in the design of molecular motors.
To understand these effects theoretically, we derived a
generalized Langer theory that takes into account con-
figuration dependent mobility matrices. We hope that
this theory will find applications also in other fields.
We thank the German Excellence Initiative for finan-
cial support via the program NIM. RN and UG are grate-
ful for the hospitality of the CTBP at UCSD, where part
of this work was done, and for financial support by the
CeNS in Munich and the DFG.
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