Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles

1.6 MB·English

by  Eva Zimmermann

Effective rates from thermodynamically consistent coarse-graining of models for
molecular motors with probe particles
Eva Zimmermann and Udo Seifert
II. Institut fu¨r Theoretische Physik, Universita¨t Stuttgart, 70550 Stuttgart, Germany
(Dated: February 2, 2015)
Many single molecule experiments for molecular motors comprise not only the motor but also
large probe particles coupled to it. The theoretical analysis of these assays, however, often takes
5
1 into account only the degrees of freedom representing the motor. We present a coarse-graining
0 method that maps a model comprising two coupled degrees of freedom which represent motor and
2 probe particle to such an effective one-particle model by eliminating the dynamics of the probe
particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey
n
a local detailed balance condition and reproduce the net currents. Moreover, the average entropy
a
productionaswellasthethermodynamicefficiencyisinvariantunderthiscoarse-grainingprocedure.
J
Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained
9
transitionratescoincidewiththetransitionratesofthetraditionallyusedone-particlemotormodels.
2
Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We

] applythis coarse-graining method tospecific case studiesof theF1-ATPaseand thekinesin motor.
h
c
e I. INTRODUCTION network description [35–37] or eliminating slow (invisi-
m ble) degrees of freedom [38–40]. It was found that, in
- Inmanysinglemoleculeexperimentsbeadsthatareat- general, coarse-graining has implications on the entropy
at tached to molecular motors are used to infer properties production and, in particular [41], dissipation. In the
t context of biological systems and especially molecular
s of the motor protein from the analysis of the trajectory
. of these probe particles. In particular, external forces motors,coarse-grainingproceduresmostlyfocusonelim-
t
a can be exerted on the motor via such a probe particle inating selected states of the motor [37, 42] or on reduc-
m ing continuous (ratchet) models to discrete-state models
[1, 2]. In the theoretical analysis of such assays,the mo-
- torisusuallymodelledasaparticlehoppingonadiscrete [43–47].
d
state space with transitions governed by a master equa-
n
tion [3–8]. Alternatively, the so called ratchet models
o
c combinecontinuousdiffusivespatialmotionwithstochas- In the present paper, we introduce a coarse-graining
[ tic switching between different potentials corresponding
procedure that allows to reduce molecular motor-bead
to different chemical states [9, 10]. These approaches of-
1 modelstoeffectiveone-particlemodelswithdiscretemo-
v tencompriseonlyoneparticleexplicitly,representingthe tor states with the external force acting directly on the
6 motor. The contribution of external forces which in the
effective motor particle. We eliminate the explicit dy-
1 experimentsactonthemotoronlyviatheprobearethen
namics ofthe probeparticle completely still maintaining
6 includedinthetransitionrates[5,6,11–17](orLangevin
the correct local detailed balance condition for the ef-
7
equationfor the spatialcoordinate[18, 19])of the motor
0 fective motortransitionratesandpreservingthe average
particle directly. However, theoretical models that are
. currentsofthesystem. Asamainresult,wefindthatthe
1 used to reproduce the experimental observations should
coarse-grained rates show a more complex force depen-
0 comprise at least two (coupled) degrees of freedom, one
dence than the usually assumed exponential behaviour
5
forthemotorandonefortheprobeparticle. Suchmodels
1 andamorecomplexconcentrationdependencethanmass
consistingofonedegreeoffreedomhoppingonadiscrete
: action law kinetics.
v state space representing the motor coupled to a contin-
Xi uously moving degree of freedom representing the probe
arediscussedin[20–27]. While multi-particlemodelsare
r
a more precise and better represent the actual experimen- The paper is organized as follows. In section II, we
talsetup,one-particlemodelsarewidely-usedtoymodels introduce our coarse-graining method on the basis of a
often applied to illustrate basic ideas. simple motor-beadmodel with only one motor state and
Simplifying the description of systems consisting of applyittoamodelfortheF -ATPase[26]. InsectionIII,
1
many degrees of freedom with a concomitant large state wegeneralizetheproceduretomotormodelswithseveral
spacewhilestillmaintainingimportantpropertiesiscom- internal states and apply it to both a refined model for
monly known as coarse-graining. In the context of theF -ATPaseandtoakinesinmodel. Apossibleexperi-
1
stochastic thermodynamics [28], various coarse-graining mentalimplementationofourmethodispresentedinsec-
methodshavebeenapplied,e.g.,lumpingtogetherstates tionIV.We showthat entropy productionandefficiency
ofadiscretestatespaceamongwhichtransitionsarefast remaininvariant under this coarse-grainingprocedure in
[29–32], averaging over states for discrete [33] or contin- section V, discuss implications on the stall conditions in
uous processes [32, 34], eliminating single states from a section VI and conclude in section VII.
2
II. GENERAL ONE-STATE MOTOR MODEL
A. Explicit motor-bead dynamics
The general model for motor proteins with only one
chemical state consists of one degree of freedom repre-
senting the motor which jumps between discrete states
n(t) separated by a distance d. The motor is coupled
withtheseconddegreeoffreedomrepresentingtheprobe
particle via some kind of elastic linker, see Fig. 1 [26].
The motion of the probe particle with continuous co-
ordinate x(t) is described by an overdamped Langevin
FIG. 1. (Color online) Schematic representation of a motor-
equationwithfrictioncoefficientγ andconstantexternal
beadmodelcomprisingaone-statemotor(smallbluesphere)
force f ,
ex attached via an elastic linker to theprobe particle (large red
sphere). An external force fex is applied to the bead. The
x˙(t)=(−∂xV(n−x)−fex)/γ+ζ(t), (1) transition rates of the motor are denoted by w+(n,x) and
w−(n,x). The load sharing factors θ+ and θ− indicate the
including the potential energy of the linker V(n − x)
positionofanunderlyingunresolvedpotentialbarrierrelative
and thermal noise ζ(t) with correlations hζ(t )ζ(t )i =
2 1 to theminimum of thefree energy landscape of themotor.
2δ(t −t )/γ. Throughout the paper, we set k T = 1.
2 1 B
This choice implies that the product of force f and
ex
distance d appearing in the figures below is measured particle hopping between states separated by d. We
in units of k T. The (instantaneous) distance between thus have to eliminate the x-coordinate from the (n,x)-
B
motor and probe is denoted by y. The system is charac- description resulting in a system characterized only by
terizedbythepairofvariables(n,x)andis“bipartite”in n.
these variables since transitions do not happen in both For the coarse-grainedmodel, we impose the following
variables at the same time. The transition rates of the conditions. Thecoarse-grainedtransitionratesΩ±which
motor fulfill a local detailed balance (LDB) condition advance the effective particle by d should obey a LDB
condition
w+(y)
=exp[∆µ−V(y+d)+V(y)]. (2) Ω+
w−(y+d) =exp[∆µ−f d] (5)
Ω− ex
Thefreeenergychangeofthesolvent∆µ≡µ −µ −µ
T D P astheforceisnowassumedtoactdirectlyontheeffective
with µ =µeq+ln(c /ceq) and nucleotide concentrations
i i i i motorparticle. Furthermore,we requirethat the coarse-
c isassociatedwithATPturnover. Theprobabilityden-
i grained particle moves with the same average velocity
sity p(y) for the distance y obeys a Fokker-Planck-type
in the steady state as the motor and the probe in the
equation
original model, i.e.,
∂tp(y)=∂y((∂yV(y)−fex) p(y)+∂yp(y))/γ v =d(Ω+−Ω−). (6)
+w+(y−d)p(y−d)+w−(y+d)p(y+d)
Solving the linear system of equations (5, 6) yields the
− w+(y)+w−(y) p(y). (3)
coarse-grainedrates
(cid:0) (cid:1)
For constant nucleotide concentrations, the system vexp[∆µ−f d]/d
Ω+ = ex (7)
reaches a non-equilibrium stationary state (NESS) with
exp[∆µ−f d]−1
ex
constant average velocity
v/d
−
Ω = . (8)
∞
exp[∆µ−f d]−1
v ≡d ps(y)(w+(y)−w−(y))dy (4) ex
Z
−∞ Thecoarse-grainedratescanbeinterpretedaseffective
∞
= ps(y)(∂ V(y)−f )/γdy transition rates that correspond to a transition process
Z y ex after which both particles, motor and probe, have ad-
−∞
vanced a distance ±d. In principle, there are (for any y)
and stationary distribution ps(y). many possible displacement processes to advance both
particles by d, including ones with l forward and l −1
backward motor jumps. The coarse-grained rate corre-
B. Coarse-graining procedure spondstotheratewithwhichonesucheffectivedisplace-
ment will happen.
Inthecoarse-graineddescriptionofthemodelwewant In general, the coarse-grainedrates depend (via v) on
to map the motor-bead system to one effective motor allmodelparameters,including the frictioncoefficientof
3
theprobeparticleandthespecificpotentialofthelinker. This expression inserted into eqs. (7, 8) yields
If one had chosen coarse-grainedrates by just averaging
over the positions of the probe particle, i.e., by Ωˆ+ =w eµT−fexdθ+, (15)
0
hw±i= ∞ ps(y)w±(y)dy, (9) Ωˆ− =w0eµD+µP+fexdθ− (16)
Z
−∞
independent of any specific linker potential V(y). Since
one would have obtained rates that yield the correct av- this forcedependence is purelyexponentialwiththe cor-
erage velocity but do not fulfill the LDB condition, as rect load sharing factor, these expressions represent ex-
discussed in section IIE below. actly the rates typically usedin one-particle models. We
For a more explicit analysis, we must specify the for- noticethatwithinthisapproximationΩ+ =hw+(y)iand
ward and backwardrates of the motor. We choose [26] Ω− = hw−(y)i holds true, which is in agreement with
other coarse-graining procedures in the time-scale sepa-
w+(y)=w exp[µ+−V(y+dθ+)+V(y)] (10)
0 ration limit, e.g., [31–33].
− − −
w (y)=w0exp[µ −V(y−dθ )+V(y)] (11) Note thatonly transitionratesofthe motorwhosede-
where θ+ and θ− are the load-sharing factors with pendence onthe linker potentialis chosenaccordinglyin
θ+ +θ− = 1 and µ+ = µ , µ− = µ +µ . We assume the Kramersform(eqs. (10,11)) leadgenericallyto con-
T D P sistentcoarse-grainedandaveragedrateswhenusingthe
an exponentialdependence of the transition rateson the
fast-bead limit of ps(y).
potential difference of the linker according to Kramers’
theory. Thisexponentialdependenceonthepotentialdif-
ference is similar to one-particle models where the rates
D. Example: F1-ATPase
ofthemotortypicallydependexponentiallyontheexter-
nal force with a corresponding load-sharing factor [3, 5].
Ingeneral,astrongtime-scaleseparationbetweenmo-
torandprobeisnotnecessarilyrealistic. Inthiscase,eq.
C. Time-scale separation (3) must be solved numerically. We will use the model
introduced in [26], see Fig. 1, with a harmonic poten-
tial V(y) = κy2/2 as a simple example to illustrate our
In this section, we will investigate under which condi-
coarse-grainingprocedure.
tions the coarse-grainedrates(7,8)canbe expressedus-
InFig. 2,theresultsforΩ+ andΩ− areshownforvari-
ingasingleexponentialdependenceontheexternalforce
ousvaluesofthefrictioncoefficientγ. Withdecreasingγ,
as typically assumed for mechanical transitions within
the rates approach their corresponding fast-bead limits,
one-particle models [3, 5].
Ωˆ+ and Ωˆ−. These values are upper bounds because de-
Inserting eqs. (10, 11) in eq. (3) in the NESS shows
creasingγ impliessmallerprobeparticleswhichexertless
that the contribution due to motor jumps is weighted
drag on the motor. For finite γ, the coarse-grainedrates
with a (dimensionless) prefactor
do notshow a single exponentialdependence onf over
ex
ε≡w exp[µeq]d2γ. (12) the whole range of external forces. Such a dependence,
0 T
however, is usually assumed to hold within one-particle
Here, w exp[µeq] determines the timescale of the transi-
0 T models. Moreover,thecoarse-grainedratesdependonγ,
tions of the motor while γd2 determines the timescale of
whichisaparameternotincorporatedexplicitlyinmany
the dynamics of the probe particle. The latter is mainly
one-particle models.
governedby the size of the bead and the step size of the
The experimentally accessible values of γ covera wide
motor whereasw exp[µeq] is determined by the attempt
0 T range of the values chosen in Fig. 2. A dimer of
frequency andalsoby the absolutenucleotide concentra-
polystyrene beads (≃ 280 nm) as used in [49–52] cor-
tions. responds to γ = 0.5s/d2 (red (dark gray) line with
If the dynamics of the bead is much faster than the
squares) while a 40nm-gold particle [52–54] corresponds
transitionsofthemotor,time-scaleseparationholdswith to γ = 5 · 10−4s/d2 (yellow (light gray) line with tri-
ε → 0 [31, 48]. In this limit of fast bead relaxation,
angles). Especially for large external forces, the coarse-
denoted throughout by a caret, the stationary solution
grainedratesdeviate stronglyfromtheirasymptotic val-
of eq. (3) in the NESS becomes
ues even for a probe as small as the gold particle.
pˆs(y)=exp[−V(y)+f y]/N (13) The average velocity as shown in Fig. 2 also strongly
ex
depends on the friction coefficient of the probe particle,
∞
withN ≡ exp[−V(y)+f y]dy. The averageveloc- especially for large external forces. In this regime, for
−∞ ex
ity is thenRgiven by large γ, the velocity is dominated by the friction experi-
enced by the probe while for small γ the probe relaxes
∞
vˆ=d pˆs(y)(w+(y)−w−(y))dy almostimmediatelyandthevelocityisdominatedbythe
Z−∞ timescale of the motor jumps.
=dw eµT−fexdθ+ −eµD+µP+fexdθ− . (14) Another option to reach the fast-bead limit is to use
0
(cid:16) (cid:17) very smallnucleotide concentrations. In Fig. 3, we show
4
100 1010 100 100
10−5 100 10−5 10−5
10−10
10−10
10−10 10−10 10−15
0 20 40 0 20 40 0 20 40 0 20 40
200
FIG. 3. (Color online) Coarse-grained rates Ω+ and Ω− as
functionsoffexdforvariouscT,cD intherange2·10−5M ≥
0 cT,cD ≥ 2·10−12M (from topto bottom). With decreasing
cT,cD, the rates approach the fast-bead limits Ωˆ+ and Ωˆ−
(straight lines). Parameters: κ = 40d−2, γ = 0.5s/d2, cP =
−200 10−3M, ∆µ=19, θ+ =0.1, w0exp[µeTq]/ceTq =3·107(Ms)−1.
−400
In summary, we find that for the F -ATPase under
−10 0 10 20 30 40 50 1
realistic experimental conditions the rates in a coarse-
graineddescriptioncomprisingonlyoneeffectiveparticle
thatsatisfytheLDBconditioneq. (5)andreproducethe
FIG. 2. (Color online) Coarse-grained rates Ω+ and Ω−
correctaveragevelocity v cannot be writtenin the form
v(taorpio)uasnfdricatvieornagceoevffielociceintyts(bγotitnomth)easrafnugnect5iosn/sd2of≥fexγd fo≥r ofa single exponentialdependence onthe externalforce.
5 · 10−10s/d2 (from bottom to top). With decreasing γ,
the rates and the velocity approach the corresponding fast-
bead limit (solid black lines). Parameters: κ = 40d−2, E. Comparison of coarse-grained with averaged
cT = cD = 2·10−6M, cP = 10−3M, ∆µ = 19, θ+ = 0.1, rates
w0exp[µeTq]/ceTq =3·107(Ms)−1.
Instead of defining the coarse-grained rates according
to eqs. (7, 8), one might be tempted to use the averaged
the coarse-grained rates for various ATP and ADP con- rates (9) as a definition for the coarse-grained rates. In
centrations. With decreasing nucleotide concentration Fig. 4, we show the averaged rates of our F -ATPase
1
(atfixed∆µ),theratesapproachtheasymptoticΩˆ+ and model as well as their ratio corresponding to the LDB
Ωˆ−. However, it is very hard to do experiments at con- condition. Wefindthatbothhw+iandhw−i(forthelat-
centrationssmallerthan≃10−7Masjumpsofthemotor terlessvisibleintheplot)exhibitnon-monotonicdepen-
are then very rare. dence on the external force. For external forces slightly
larger than the stall force, hw+i increases with increas-
In Fig. 2 and in Fig. 3 the dependence of the coarse-
ing f due to the fact that in this region the system
grained rates on the external force exhibits two different ex
moves backward with motor jumps following the probe
regimes. Up to values of the external force of roughly
which leads to a peak at small y in ps(y). On the other
15/d, the coarse-grainedrates can be well approximated
hand, hw−i exhibits a minimum around stall conditions
byasingleexponentialdependence onf withthe same
ex
slope as in the fast-bead limit, dθ+ or dθ−, respectively. for large γ since in this region, ps(y) misses a peak at
large y ≃1.
However,forlargeγ andlargec ,eveninthisregime,the
T
absolute values of the coarse-grainedrates deviate up to A severe issue appears regarding the LDB condition.
twoordersofmagnitudefromtheirfast-beadapproxima- Thecorrespondingratiooftheaveragedratesisalsoplot-
tion. Forsuchparameters,assumingamono-exponential ted in Fig. 4 where it can be clearly seen that the LDB
dependence on f with the above slope would not be condition is not fulfilled (except in the fast-bead limit).
ex
appropriate either.
For large external forces, all coarse-grained rates de-
F. Without external force
viate significantly from their fast-bead limits. We find
again a mono-exponential decay for Ω+ but now with
slope−dwhereasΩ− growsonlylinearlywithincreasing Eventhough we have motivated this paper by empha-
f . Thissofarunaccountedforbehaviourcanbeunder- sizing that external forces are typically applied to probe
ex
stood by considering the limit f → ∞ as discussed in particles, it should be obvious that our approach holds
ex
detail in the Appendix. The crossover from one regime true for molecular motors transporting cargo subject to
to the other occurs beyond the stall force f =∆µ/d. Stokes friction in the absence of external forces.
ex
5
10−5
102 1010
105
101
100 10−10
100 100
10−10
0 20 40 0 20 40 10−8 10−6 10−4 10−2 100 10−8 10−6 10−4 10−2 100
1010
105 105
100
10−5
10−10 100
−10 0 10 20 30 40 50 10−8 10−6 10−4 10−2 100
FIG. 4. (Color online) Top: Average rates hw+i and hw−i
as functions of fexd for various γ in the range 5s/d2 ≥ γ ≥ FIG.5. (Coloronline)Coarse-grainedratesΩ+ andΩ− (top)
5·10−9s/d2. With decreasing γ, therates approach Ωˆ+, Ωˆ− and average velocity (bottom) for various γ and fex = 0
(solid black lines). Bottom: Ratio of + and − rates. In con- as functions of cT. Since cD and cP are fixed, ∆µ also
trasttoΩ+,Ω− (largereddots),theaveragedmotorratesdo increases with cT. The rates and the velocity approach
the fast-bead approximation (solid black lines). Parameters:
nteortsfaurlefiltlhtehesaLmDeBascoinndFitigio.n2.(solid black line). The parame- cw5D0·1e=0x−p9[2µseT·/q1d]0/2−c(eT6fqrMo=m,3cbP·o1t0=t7o(mM1·st)o1−0t1−o,3pγM).i,nκthe=ra4n0gde−52,s/θd+2 =≥γ0.1≥,
For one-particle models, the friction coefficient of the
probe can not be taken into account explicitly. One locity is then linear in c as ina one-particlemodel. For
T
ratherhasto incorporatethe drageffectofthe beadinto large γ, eliminating the cargo by coarse-graining yields
the motor rates [46]. If one wants to analyze experimen- coarse-grainedratesthatarenot linearinthe concentra-
tal data obtained from probe particles of different sizes, tions although the motor rates are still subject to mass
onethenhastousedifferentvaluesofthemotorratesfor action law kinetics. Moreover,the velocity then exhibits
each data set. a sub-linear dependence reminiscent of the typical satu-
For the rather dilute solutions used in experiments ration effect for large c .
T
[49, 51, 55] one generally assumes that the motor dy-
namics is subject to mass action law kinetics, i.e., that
the transition rates depend linearly on the correspond- G. Comparison of full and coarse-grained
ing concentration of nucleotides. Obviously, this linear trajectories
dependence holds for all concentrations and beads of all
sizes for one-particle models. When keeping c and c
D P Trajectories of motor and probe generated by a simu-
fixed,the averagevelocityofaone-statemotorwillshow
lationofthecompletemodeloftheF -ATPaseareshown
1
a purely linear dependence on c .
T in Fig. 6. Additionaly, Fig. 6 contains a trajectory ob-
Theexperimentalanalysisoftheaveragevelocityofthe tained from simulating the corresponding coarse-grained
F1-ATPase as function of cT (for fixed cD, cP) reveals a model. The average velocity of both models is the same
saturation of the velocity for large ATP concentrations (by definition, see eq. (6)), whereas the coarse-grained
which sets in earlier for large beads [53]. While such a model produces trajectories that are “more random”.
saturationisusuallyattributedtothehydrolysisstep,we This behavior occurs since the coarse-grained rates are
findthatasub-lineardependenceofthevelocitycanalso constant(for fixedparameters)and produce a simple bi-
be caused by the drag of the probe particle. ased random walk. The motor transition rates of the
In Fig. 5, the coarse-grained rates as well as the ve- complete model, however,depend on the actual position
locity areshownasafunction ofthe ATP concentration. oftheprobeandarethereforeimplicitlytime-dependent.
Withdecreasingγ,thecoarse-grainedratesapproachthe Sincefastsuccessivemotorjumpsaresuppressed,thetra-
fast-beadlimitandthemassactionlawkinetics. Theve- jectory of the complete model is less random [21, 56].
6
25 0
−5
20
−10
15
−15
10
−20
5
−25
0 −30
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
30 25
25
20
20
15
15
10
10
5
5
0 0 FIG. 7. (Color online) Network representation of a motor-
0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 bead model with four internal motor states and discretized
statespaceoftheprobeparticle(left). Eachrowofblackdots
representsonemotorstatewhilethedotsthemselvesrepresent
specific distances y accessible to the probe particle (via the
FIG. 6. (Color online) Trajectories of the one-state model vertical red lines) within the same motor state. Transitions
for the F1-ATPase for several parameter sets obtained from betweenmotorstateseitherleaveythesame(horizontalgreen
simulations. The trajectory of the detailed model (motor: lines)orcanadvancethemotorbydα andchangey(diagonal
ij
step-likeblue lines, probe: fluctuatingred lines) is shown to- blue lines). The top view of this network corresponds to the
gether with a trajectory of its corresponding coarse-grained coarse-grained version of this model (right).
model(green(lightgray)). Parameters: κ=40d−2,θ+ =0.1,
wcT0e=xpc[DµeT=q]/2c·eTq10−=6 3M·,1c0P7=(M0.s0)0−11,Mγ(t=op0le.5fts)/;dγ2,=f0ex.5s=/d02,, transitions the edges of a network. Transitions between
fex = 40d−1, cT = cD = 2·10−6 M, cP = 0.001 M (top the motor states i and j change the free energy by
right); γ = 0.005s/d2, fex = 0, cT = cD = 2 · 10−6 M,
McP,=cD0.=0021·M10(−b6oMtto,mcPle=ft)0;.0γ01=M0.5(bs/odtt2o,mfexri=gh0t),.cT =0.001 ∆Fiαj ≡Fj −Fi−∆µαij (17)
where F −F is the free energy difference of the inter-
j i
nal states of the motor and ∆µα = −∆µα is the free
ij ji
energy change of the solvent. Depending on the transi-
The influence of parameters like the probe size or the
tion, ∆µα is given by µ , µ , µ or any combination
ATPconcentrationonthe dynamicsisvisibleinthebot- ij T D P
thereof or 0. Transitions may also advance the motor
tompanelsofFig. 6. Whiletheaveragevelocityisalmost
a distance dα = −dα. Since we allow for several transi-
the same, the trajectories of the complete model differ ij ji
tionsconnectingtwostates,weassignanadditionalindex
significantly. Using a small probe with a small friction
α to the transitions indicating which link between i and
coefficient, the probe relaxes to the potential minimum
j is used. An example for the network of a full system
ofthe linkerbeforethenextmotorjumpoccurs,whereas
comprising motor and probe particle is shown in Fig. 7,
the large probe cannotrelax [25]. LargeATP concentra-
wherethestatespaceoftheprobeisdiscretizedforbetter
tions induce many forward and succesive backward mo-
presentation.
tor jumps that are absent at lower ATP concentrations.
The Fokker-Planck-type equation for such models is
These details are not capturedin the coarse-grainedtra-
given by
jectories.
∂ p (y)=∂ ((∂ V(y)−f )p (y)+∂ p (y))/γ
t i y y ex i y i
+ wα(y+dα)p (y+dα)−wα(y)p (y). (18)
III. MOTOR MODELS WITH SEVERAL ji ij j ij ij i
X
INTERNAL STATES j,α
with transitionrates of the motor that obey a LDB con-
A. Explicit motor-bead dynamics and dition
coarse-graining procedure wα(y)
ij =exp[−∆Fα−V(y+dα)+V(y)]. (19)
wα(y+dα) ij ij
Inthissection,wewillgeneralizethemodeltakinginto ji ij
account several different internal states of the motor la- Thecoarse-grainedversionofsuchamodelshouldtake
belledbyi. Themotorstatesrepresentthenodesandthe into account the different states of the motor as well as
7
the severalpossibleα-transitionsbetweeniandj. Thus, Transitionswithratesdependingony butwithdα =0
ij
themotornetwork(includingallmotorcycles)shouldbe have coarse-grained rates that depend on f only im-
ex
conservedunder coarse-graining. To accountforthe sev- plicitely via jα and P as will be discussed below in
ij i,j
eral internal states, we require that the coarse-grained section IIID for the chemical transition rates of kinesin.
rates should obey a LDB condition and the operational TheratesdeterminedfromtheLDBconditioneq. (20),
current[57] from motor state i to motor state j via edge thepopulationsP andtheoperationalcurrentsarealge-
i
α should be conserved. The operational current is the braically consistent with the fact that a full set of rates
sum over all y-dependent net transition currents that Ωα will uniquely determine the populations P on the
ij i
contribute to the transition i → j. Conserving the op- coarse-grained network. Consistency can be seen by in-
erational currents corresponds to the condition of repro- tegratingtheFokker-Planckequation(18)overyyielding
ducingthecorrectmeanvelocityfortheone-statemodel. the coarse-grainedmaster equation
The above conditions read
∂ P = jα = P Ωα −P Ωα, (28)
t i ji j ji i ij
X X
Ωα j,α j,α
ij =exp[−∆Fα−f dα] (20)
Ωαji ij ex ij whose stationary solution in the NESS can be expressed
as a function of the rates Ωα [57, 58]. Thus, the expres-
and ij
sionofanycurrentobservableintermsoftheoperational
P Ωα −P Ωα =jα (21) currentsisconsistentwithitsexpressionintermsofcycle
i ij j ji ij
currents on the coarse-grainednetwork.
with the operational current
∞
B. Time-scale separation
jα ≡ p (y)wα(y)−p (y+dα)wα(y+dα) dy
ij Z i ij j ij ji ij
(cid:2) (cid:3)
−∞
Similar to the one-state-model, we explore the conse-
=−jα (22) quences of a putative time-scale separation between the
ji
dynamics of motor and probe for each motor transition.
and the marginal distribution
In the limit γ → 0 (formally equivalent to ε → 0 but
∞ hereonewouldhaveseveralε withintheFokker-Planck
ij
P = p (y)dy. (23)
i Z i equation and all go to 0) the solution of eq. (18) in the
−∞
NESS becomes, analogously to [29, 32],
TheseequationscanbesolvedforΩα andΩα usingsim-
ij ji
ple algebra which yields the rates pˆs(y)=Pˆ exp[−V(y)+f y]/N. (29)
i i ex
exp[−∆Fα−f dα]
Ωα =jα ij ex ij (24) Themarginaldistributioncanbeobtainedusingeq. (18)
ij ij Piexp[−∆Fiαj −fexdαij]−Pj with its solution for fast bead relaxation
1 ∞
Ωαji =jiαj Piexp[−∆Fiαj −fexdαij]−Pj. (25) ∂tPˆi =Z−∞∂tpˆsi(y)dy
Inprinciple,itis sufficientto use onlyeq. (24),since Ωαji = Pˆjhwjαiiy −Pˆihwiαjiy = ˆjjαi =0. (30)
takes exactly this form with jα = −jα,∆Fα = −∆Fα X(cid:16) (cid:17) X
ij ji ij ji j,α j,α
and dα = −dα. This equivalent procedure would be
ij ji
moresymmetricandtreatalltransitionratesonanequal For Kramers-type transition rates like eqs. (10, 11),
footingbuttheLDBconditionisthenlessobvious. Note
that without the LDB condition (20), the stated condi- wα(y)=kα exp[µα,+−V(y+dαθα,+)+V(y)] (31)
ij ij ij ij ij
tionsofPi andjiαj wouldalsobe compatible withcoarse- wα(y)=kα exp[µα,−−V(y−dαθα,−)+V(y)], (32)
grained rates like the ones in, e.g., [31, 33]. ji ji ij ij ij
Transitions whose rates are independent of the linker with µα,+ −µα,− = ∆µα and kα/kα = exp[−F +F ],
elongation y and hence have dα = 0 retrieve their orig- ij ij ij ij ji j i
ij the y-averagedrates hwαi and hwαi become
inal rate constants through this coarse-graining proce- ij y ji y
dure. For such a transition, jα is given by
ij hwαi =kα exp[µα,+−f dαθα,+] (33)
ij y ij ij ex ij ij
jiαj =Piwiαj −Pjwjαi (26) hwαi =kα exp[µα,−+f dαθα,−]. (34)
ji y ji ij ex ij ij
with rates fulfilling the LDB condition wα/wα =
exp[−∆Fα]. Inserting jα in eqs. (24, 25) aijnd juising Thechangeofchemicalfreeenergy∆µα issplitintoµα,+
ij ij ij ij
the LDB condition and dα =0 immediately yields andµα,− indicatingthatbothdirectionsofthetransition
ij ij
can involve binding and release of the chemical species
Ωαij =wiαj, Ωαji =wjαi. (27) that account for ∆µα. The free energy change arising
ij
8
from changing the motor state, Fj −Fi, is incorporated 1010
intheattemptfrequencieskα ofthecorrespondingstates.
ij
Inserting the operational current in the form of eq. (30) 100
withtheseaveragedrates,simplecalculusshowsthatthe
coarse-grainedrates (24) and (25) reduce to 100
Ωˆα =kα exp[µα,+−f dαθα,+] (35) 10−5
ij ij ij ex ij ij
Ωˆαji =kjαiexp[µαij,−+fexdαijθiαj,−] (36) 10−10
−20 0 20 40−20 0 20 40
which is again consistent with transition rates of one-
102
particle models thatassume a purelyexponentialdepen- 103
dence on the external force.
100
102
10−2
C. Example: F1-ATPase with intermediate step 101
10−4
1. With external force
100
−20 0 20 40−20 0 20 40
The 120◦ stepofthe F -ATPaseisknowntoconsistof
1
a 90◦ and a 30◦ substep [53]. Such a stepping behavior
200
canbemodelledwithaunicyclicmotorwithtwointernal
states. A schematic representation of a system compris-
ingaprobeparticleandamotorwithtwointernalstates 0
is shown in Fig. 8. The two different pathways for tran-
◦
sitions between the states 1 and 2 correspond to the 90
and 30◦ substeps of the F1-ATPase, respectively. −200
LikeinsectionIIDfortheone-statemodel,weexamine
thecoarse-grainedratesforthe90◦and30◦ stepsandthe
velocity which are shown in Fig. 9. Similar to the 120◦- −20 −10 0 10 20 30 40
scenario, the rates approach their fast-bead limit with
decreasing γ.
Asintheone-stepmodel,thedependenceofthecoarse- FIG.9. (Color online) Coarse-grained rates for the90◦ (top)
grained rates on the external force shows two regimes. and the 30◦ (center) substep and average velocity (bottom)
For small external forces, the rates can be well approxi- as functions of fexd for various γ in the range 5s/d2 ≥ γ ≥
5·10−10s/d2 (from bottom to top). With decreasing γ, the
mated by a single exponential dependence on f with
ex
slope ±dαθα,± in most cases. For large probe parti- ratesandthevelocityapproachtheircorrespondingfast-bead
cnloers,shhoowweijmveoirjn,oth-eexrpaotneesnnteiaitlhdeerpmenadtcehncteheonabfseoxluwteithvatlhuee l21i0m·71i(t0M−(ss6)oM−li1d,,cbkPl9a0=cekx1pl0i[n−µe3esqM)]./,cPeθqa9+r0=a,3m03e6=t6e7r0.s.5:1(,Mκks=19)20−4e1x0,dpk−[µ320eT,q=c]/Tc1eT=0q00=csD−3=1·,
above slope. For large forces, the forward rates decay k30exp[µeq]/c2e1q = 40D(MsD)−1. The attempt fr2e1quencies are
12 P P
chosenonthebasisof[53,54]whereverysmallprobeparticles
havebeen used.
fasterwhereasthebackwardratesgrowmoreslowlythan
in the fast-bead limit.
Concerning the average velocity, strong deviations
from the fast-bead limit occur only for the largest fric-
tion coefficients. Using small beads, the force-velocity
relationresulting fromour coarse-grainingprocedureco-
incides well with the one obtained from a one-particle
model due to the fact that the velocity involvesonly dif-
FIG. 8. (Color online) Schematic representation of a motor- ferences of the rates multiplied with the marginal dis-
bead model for theF1-ATPasewith twoinernal states of the tribution rather than the rates themselves. For large
motor, 1 (blue (dark gray)) and 2 (pale green (light gray)). external forces and small γ, the velocity is significantly
Transition between states 1 to 2 corresponding to the 90◦ smaller than in the one-state model since the motor has
◦
(30 )substeparelabelledwithsuperscript90(30). Thetran- to take two successive steps to cover the full d. The
sition rates are chosen accordingly to eqs. (31, 32).
force-velocity relations for the two-state as well as for
9
the one-state model reproduce very well the exerimen- 100
tally determined force-velocity relation from [51] for the
corresponding value of the friction coefficient γ. 105
The limiting cases f → ±∞ are more involved here
ex
than in the one-state model since one has to account 10−5
for the dependence of the P ’s on the external force.
i
However, as long as the Pj’s do not decay faster than 100
exp[−f dα], it is still possible to approximate the rates
ex ij
(24, 25) by
10−8 10−6 10−4 10−2 100 10−8 10−6 10−4 10−2 100
10−1
Ωα ≈−jαexp[−∆Fα−f dα]/P , (37)
ij ij ij ex ij j 103
Ωα ≈−jα/P (38)
ji ij j
10−2
since P is bounded by 1.
i 102
For the F -ATPase model, the numerical analysis in
1
the f →∞ limit yields a linear dependence of hyi and
ex
jα on f . We also find that P decays exponentially 10−3
ij ex 2
while P approaches 1. Hence, Ω90 and Ω30 decay expo- 101
1 12 21 10−8 10−6 10−4 10−2 100 10−8 10−6 10−4 10−2 100
nentially with slope −d90 =−0.75d and −d30 =−0.25d,
12 21
respectively, like in the one-state model but Ω90 now
21
grows exponentially with a smaller exponent while Ω30
12
still grows linearly.
102
2. Without external force
100
Just as for the one-state-model, we examine the de-
pendenceofthecoarse-grainedratesontheATPconcen-
10−2
tration in the absence of external forces. 10−8 10−6 10−4 10−2 100
◦
Fig. 10 shows the coarse-grainedrates for the 90 and
the 30◦ substeps as well as the avergage velocity. With
decreasingγ,thecoarse-grainedratesapproachthemass ◦
FIG.10. (Coloronline)Coarse-grainedratesforthe90 (top)
action law kinetics for the corresponding one-particle ◦
and the 30 (center) substep and average velocity (bottom)
rates. In contrast to the one-state model, even in this
for various γ and fex = 0 as functions of cT. Since cD and
limit, the velocity shows saturation. This is due to the cP are fixed, ∆µ also increases with cT. The rates and the
fact that the timescale of the hydrolysis reaction is in- velocity approach the fast-bead approximation (solid black
dependent of the ATP concentration and represents the lines). Parameters: cD = 2·10−6M, cP = 1·10−3M, κ =
limitingeffectforthevelocity. Thedependenceoftheav- 40d−2,θ+ =0.1,γintherange5s/d2 ≥γ ≥ 5·10−10s/d2
90,30
erage velocity on the ATP concentration is reminiscent (from bottom to top).
of a Michaelis-Menten kinetics and coincides well with
experimenal results for several different probe particles
as shown in [53].
For large beads, the coarse-graining process yields
rates that are no longer linear in the corresponding con-
centrations. In this regime,the sub-lineardependence of rate depends only weakly on γ for small ATP concen-
the velocity on the ATP concentration appears already trations which is reminiscent of the experimental obser-
for smaller ATP concentrations. Comparing the velocity vation that the ATP binding rate to the motor depends
curves of the two-state model with the one-state model, only weakly on the size of the probe [52]. However, for
we find that for large beads the velocity curves almost large ATP concentrations that were not investigated in
coincide since in this regime the limiting effect for the the experiment, the 90◦ rate shows a strong dependence
velocity is the friction experienced by the bead. Thus, on γ. This is due to the fact that for small ATP concen-
using large probe particles it is not possible to infer the trations the relaxation times of all probe particles are in
underlying motor dynamics from the characteristics of the order of, or even faster than, the motor jump rates.
the velocity as a function of the ATP concentration[25]. The results for the 30◦ rate are consistent with exper-
Fig. 11 shows the coarse-grained forward rates for imental results for the hydrolysis rate [52]. Increasing
three different nucleotide concentrations and for various c decreases the P release rate in the experiment as it
P i
γ chosenas in the experiment [52]. We find that the 90◦ decreases the 30◦ rate here.
10
103
104
103
102
102
101
I II III I II III
FIG.11. (Coloronline)Coarse-grained forward ratesforvar-
ious γ = 2.1s/d2, 0.26s/d2, 0.14s/d2, 0.017s/d2, 0.003s/d2,
0.001s/d2,3.8·10−4s/d2 (frombottomtotop)fortheparam-
FIG. 12. (Color online) 6-state-model representing a kinesin
eter sets I: cT =430 nM, cP =1 nM; II: cT =1 mM, cP =1 motoradapted from [5]. Thetransition between states2and
nM; III: cT =1 mM, cP =200 mM as used in [52]. Parame- 5 is purely mechanical and corresponds to a step of length
tersthatarethesameforallsetsI-III:cD =1nM,κ=40d−2, d whereas all other transitions are purechemical transitions.
θ+ =0.1andtheattemptfrequencieskα asgiveninFig9.
90,30 ij The motor model includes three cycles: F which, in the +
The values of cD =cP =1 nM are a rough estimate because direction, includes ATP hydrolysis and forward stepping, B
thereis noinformation about these concentrations in [52].
which includes ATP hydrolysis and backward stepping in its
+directionandapurechemicalcycle(aroundthecircle)that
includes hydrolysis/synthesis of two ATP.
D. Example: Kinesin
1010 104
Asafinalmorecomplexexample,weapplyourcoarse-
102
grainingmethodtoamodelwithamulti-statemotor. We
choose the well-studied 6-state-model representing a ki- 105 100
nesinmotorintroducedin[5],see Fig. 12. Implementing
10−2
the probe particle and an elastic linker V(y), we adopt
thetransitionratesofthemotorfrom[5]andreplacethe 100 10−4
dependence on the external force by the dependence on
10−6
the elongation of the linker,
−10 0 10 20 30 −10 0 10 20 30
w+(y)=k exp[−V(y+dθ+)+V(y)], (39)
25 25
− −
w (y)=k exp[−V(y−dθ )+V(y)], (40)
52 52
FIG. 13. (Color online) Coarse-grained rates for the me-
wi+j,chem =kij1+e2xpex[∂p[Vµ+i(jy])χ ], (41) cthheanricaanlgetra0n.0s7it7ios/nds2(w≥ithγy-≥dep7e.7nd·en10ce−)10fso/rd2va(rfirooums γboitn-
y ij
tom to top). The rates approach the one-particle rates
−
− 2exp[µij] from [5] (solid black lines). Parameters: κ = 10d−2 [55],
wji,chem =kji1+exp[∂yV(y)χij]. (42) cT = 0.001M, cD = cP = 10−9M (estimated), θ+ =
The first two rates belong to the mechanical transition, 02.6·51,0χ6i(jM=s)−01.2,5,k02.115,=k12ke2x3p[µ=eTq]/kc3eT4q ==k45k5e6xp[=µeTq]k/6c1eTq ==
twlihnhekicelhrowdweeirpthetnwadocorhnaetmtehsiecarienlpsltroaeasndet-nastnheatorhuiensgcfohfraecmcetioceraxleχrttre,adnsbseiyetiot[5hn]es. 1k304·301(e0sx5)p−([s1µ),−ePqk1]3,/2ckeeP5xq2p==[µ0eDkq.2]1/64c(eeDsxq)p−=[1µ,kePqk6]55/4ecx=ePqp[(µ=keD5q22]//k·c2eD15q0)=42(kM221·s.1)−041(,Mks2)5−=1,
ij
±
The change of chemical free energy µ =µ ,µ ,µ de-
ij T D P
pends on which transition involves binding of the corre-
sponding nucleotide. We choose again V(y)=κy2/2. However,theaveragevelocity(obtainedfromourcoarse-
The coarse-grainedrates for the mechanicaltransition grained rates) as function of the external force coincides
are shown in Fig. 13. With decreasing γ, the rates verywellforalmostallγ withthevelocitycurveobtained
approach their fast-bead limit which corresponds to the from the bare motor model, see Fig. 14. Like for the
ratesusedin[5]whilestrongdeviationsoccurforfiniteγ F1-ATPase model discussed in section IIIC, this agree-
especiallyforassistingexternalforces. Thefrictioncoeffi- mentisduetothefactthatthevelocityinvolvesonlythe
cientofaprobeofsize500nmasin[55]canbecalculated difference of the rates multiplied with the marginal dis-
using Stokes’ law yielding γ ≃ 7.7·10−5d2/s. For fric- tribution. If one investigates only force-velocity curves,
tioncoefficientsinthisrange(lightgreen(lightgray)line the discrepancies between the coarse-grained rates and
with triangles), our coarse-grained rates show a distinct the one-particle rates are hardly visible.
deviation from the one-particle rates (solid black lines). In contrast to the coarse-grained rates of the F -
1