An inchworm processive mechanism is proposed to explain the motion of dimeric
molecular motors such as kinesin. We present here preliminary results for this
mechanism focusing on observables like mean velocity, coupling ratio and
efficiency versus ATP concentration and the external load F.Comment: 6 pages, 2 figure

A. Ciudad

A.M. Lacasta

Asbury

Ciudad

Fogedby

J.M. Sancho

Klumpp

Mogilner

Stratopoulos

Visscher

Physica A Statistical Mechanics and its Applications

English

arXiv

Crossref

An inchworm processive mechanism is proposed to explain the motion of dimeric molecular

motors such as kinesin.We present here preliminary results for this mechanism focusing on observables like mean velocity, coupling ratio and efficiency versus ATP concentration and the external load F.Peer Reviewe

Ciudad Álvarez, Aleix

Sancho, Jose Maria

Lacasta Palacio, Ana María

UPCommons

Dynamics of an inchworm nano-walker

An inchworm processive mechanism is proposed to explain the motion of dimeric molecular
motors such as kinesin.We present here preliminary results for this mechanism focusing on observables like mean velocity, coupling ratio and efficiency versus ATP concentration and the external load F.Peer ReviewedPreprin

Alva, Aleix

UPCommons. Portal del coneixement obert de la UPC

Dynamics of an inchworm nano–walkerA. Ciudad†‡, J.M. Sancho† and A.M. Lacasta‡† Departament d’Estructura i Constituents de la Mate`ria,Facultat de F´ısica, Universitat de Barcelona,Diagonal 647, E-08028 Barcelona, Spainand‡ Departament de F´ısica Aplicada,Universitat Polite`cnica de Catalunya,Av.Dr. Maran˜o´n 44, E-08028 Barcelona, SpainAn inchworm processive mechanism is proposed to explain the motion of dimeric molecularmotors such as kinesin.We present here preliminary results for this mechanism focusing onobservables like mean velocity, coupling ratio and efficiency versus ATP concentration andthe external load F .Molecular motor proteins transform the energy of ATP hydrolysis into mechanical work per-forming discrete steps along a periodic track. The experimental work on protein motors [1, 2, 3] hasstimulated a wide variety of modelizations, most of them based in ratchet–like potentials. The twomain candidates for the walking mechanisms of dimeric motors were inchworm or hand–over–hand.In the first case it is assumed that the first leading head advances one step which is followed instan-taneously by the trailing head. In the hand–over–hand mechanism the second head advances twosteps overpassing the first head. In Ref. [2] some experimental evidence was presented which seemto support the inchworm mechanism. Nevertheless, more precise experiments show that myosin–Vwalk in a hand–over–hand way [3]. These two different mechanisms imply different conformationalchanges in the protein structure during ATP hydrolysis. Moreover they imply a different responsewith respect the experimental control parameters [ATP] and F .Although it is commonly accepted now that some processive members of kinesin, myosin anddynein families seem to walk in a hand-over-hand fashion, it is still worth analyzing the inchwormmechanism, which could hold for other type of motors. For these reasons, we will present here avery simple model walking in a inchworm fashion with parameter values in the biological scale. Wewill also focus on some implications with experimental relevance.‡ To whom correspondence should be addressed (aciudad@ecm.ub.es)It was showed in Ref.[1] that kinesin uses a single ATP molecule to perform each step. Such relationis called the coupling ratio, which for low external loads is about 1. The temporal distribution ofthese steps is random due to the ATP diffusion until it reaches the motor. After binding thenucleotide, hydrolysis and the consequent conformational change take place displacing the wholemotor a certain distance which is usually equal to the periodicity of the track. All this process canoccur even in the presence of an opposing external force F and at low ATP concentration, althoughboth regimes decrease the mean velocity.The inchworm walking mechanism can be modeled as two linearly-coupled particles interacting witha ratchet potential. The conformational cycle is introduced as a stretching and posterior relaxingof the coupling spring. This way of modeling was introduced in Ref.[4] showing that thermalfluctuations are not strictly necessary in order to achieve the motion. Other works [5, 6, 7, 8, 9]are also based on this approach. The main difference between them is the way they model themechanical changes under the input of chemical energy. While [4] considers the conformationalchange as an increase of the equilibrium length of the spring, other literature introduce asymmetricfrictions or switches on the ratchet potential. Here we will explore a different and simple way whichallows to control with precision the amount of the input energy. Furthermore, we apply a kineticmethodology[10] based on enzymatic inhibition to get an analytical expression for the velocity as afunction of the ATP concentration and the external force F . Finally we analyze the coupling ratioand the efficiency at different values of F .We will consider the motor as two particles coupled by a spring. The set of equations in theoverdamped limit are,λx˙1 = −V′(x1)− k(x1 − x2 − L)− fs(t)−F2+ ξ1(t)λx˙2 = −V′(x2) + k(x1 − x2 − L) + fs(t)−F2+ ξ2(t), (1)where x1,x2 are the position of the trailing and the leading head, respectively. k is the stiffness ofthe harmonic spring with equilibrium length L, which is also the periodicity of the ratchet potentialV (x). Such potential has an asymmetric factor α and a barrier height V0 (See Fig.1). fs(t) is therandom chemical force and F is the external load. The thermal force is emulated through a zeromean Gaussian-white noise with a correlation, 〈ξi(t)ξi(t′)〉 = 2λkBTδ(t − t′). λ is the friction andkBT is the thermal energy. Then, at thermal equilibrium the two particles will lay, most of the time,on two consecutive potential minima. We assume that at random intervals of time, an energeticnucleotide like ATP will bind the motor and a stretching force fs will act on the system until thetotal length of the motor will be doubled, i.e. x2 − x1 = 2L. If E is the hydrolysis energy of theFIG. 1: a) Scheme of the ratchet potential the positions of the leading head (black particle) and the trailinghead (gray particle) at the three stages of the motor. i) Rest configuration. ii) The motor at the end of thestretching. iii) Final stage. b) Simulated mean velocities 〈v〉 of the center of mass versus p for F = 0, 1, 3 pNdrawn with circles, squares and triangles, respectively. Dashed lines are Michaelis-Menten fits. The resultingkinetic parameters are shown in the table.molecule and L is the displacement that it is performed, then we take the chemical force fs = E/L.The values of the parameters have been chosen in a nano scale to mimic some molecular motorssuch the kinesin. The periodicity of the potential L is taken to be the periodicity of microtubules,8nm. The asymmetric factor α = 0.8 and V0 = 50pNnm optimizes the efficiency of our model.E = 100pNnm corresponds to an accepted value for the energy of hydrolysis of an ATP andthermal energy is kBT = 4.1pNnm. The stiffness of the motor is chosen k = 1pN/nm and the dragforce λ = 2 · 10−4pNs/nm. When the motor is free from ATP, p ∈ (0, 1) is the uniform probabilityper time step ∆t of binding one molecule. When it occurs, more ATP binding is forbidden andstretching takes place until the elongation is 2L. Then, the stretching force disappears and thespring relaxes. When x2 − x1 is again L, one cycle is completed and ATP binding is allowed. Inthe absence of external load, this mechano-chemical cycle induces a L displacement of the motortowards one end of the potential. Fig. 1a shows the scheme of the process. On the other hand,our approach controls how much energy ET is applied to the system by simply multiplying Eby the number n of ATP consumed: ET = nE. The mean velocity of the motor, when the ATPconcentration is saturant and F = 0, is maximum and dependent only on the intrinsic propertiesof the motor and by E. Let ton be the time spent to perform a single step, i.e. the stretching plusthe relaxing time. Thus, Vmax = L/ton. Using the given values of the parameters, simulations showthat ton ∼ 0.012s, which gives Vmax ∼ 667nm/s. However, the global speed 〈v〉 will be sloweddown when the ATP concentration decreases. Typically, the [ATP ]-dependence on 〈v〉 is givenby the Michaelis-Menten relation [10]. In our model, we have previously defined p as the uniformprobability to get an ATP per time step ∆t and with p = 0 while the motor stretches and relaxes.It can be accepted that, as the reaction frequency is proportional to [ATP ], and then [ATP ] isproportional to p. Then, we have〈v〉 = VmaxpKM + p, (2)where KM is the Michaelis constant for the probability. From now on, we will deal with p and notwith [ATP ]. Fig.1b shows how the michaelian behavior fits well the simulated values of the meanvelocity. However, for finite values of F , both kinetic parameters Vmax and KM change. In Ref.[10]it is shown that the effect of the external load in kinesin can be interpreted as an inhibition process.This introduces a F -dependence on the two kinetic parametersVmax(F ) =Vmax(F = 0)1 + 1Kiu(FS/F−1)KM (F ) = KM (F = 0)1 + 1Kic(FS/F−1)1 + 1Kiu(FS/F−1). (3)and allows to express the velocity of the motor as a function of the two control variables p and F .FS is the stall force, i.e. the maximum load that the motor is able to carry. In our simulations,FS ∼ 5.25pN. Kiu andKic are the uncompetitive and competitive inhibition constants, respectively,and are a quantitative measure of how F affects the motor when it is free from nucleotide (Kic) orwhen it has an ATP (Kiu). From table values of Fig.1b we fit the values of the inhibition constantsobtaining Kiu ∼ 0.338 · 10−6 and Kic ∼ 2.131 · 10−6. As they are dissociation constants, the effectof the load on the ATP-bound state is greater than in the ATP-free configuration. This means thatthe force acts as an uncompetitive mixed inhibitor, while in Ref.[10] it is shown that kinesin is alsomixed but competitive. This difference is responsible of the curvature on 〈v〉 − F curves at highATP concentration. Figure 2a shows these curves with the simulation data and the predictions ofthe analytical expression with an excellent agreement.Finally, it is interesting to define the coupling ratio and the efficiency and to see how are theymodified by F . The coupling ratio r can be expressed as the quotient between the total number ofperformed steps and the total number of consumed ATP’s. On the other hand, the efficiency η canbe defined as the ratio between the work performed against F , W , and the total input of energy,nE. Thus,r ≡xCMnLη ≡WnE, (4)FIG. 2: a) Simulated mean velocities versus F for p = 1 (circles) and p = 10−5 (triangles). The insets showVmax(F ) and KM (F ) versus F . Triangles are the values from the table in 1b). Solid lines are the fits of(3) in order to get Kiu and Kic. With the two inhibition constants, (2) can be plotted obtaining the solidlines of the main figure. b) Coupling ratio r and efficiency η. Circles are simulated data and solid lines arepredictions of (5).where xCM =12(x1 + x2) and supposing that xCM (t = 0) = 0. W = FxCM , so we can writeη = rFL/E. This means that the global efficiency is simply the efficiency in a single step multipliedby the coupling ratio. We can go further if we consider (3) and the fact that Vmax is proportionalto r, and then,r =11 + 1Kiu(FS/F−1)η =LEF(1 + FKiu(Fs−F )). (5)Fig.2b shows the simulated data for r(F ) and η(F ) as well as the theoretical predictions. It isinteresting to remark that the maximum efficiency is slightly below 0.15.We have presented an inchworm mechanism which is able to perform directed transport andanalyzed how it behaves under two variables, the ATP concentration through the probability pand the external load F . The motor can be described as a uncompetitive mixed inhibitor obtaininganalytical expressions for the mean velocity as a function of the two control variables that fitsaccurately the simulated data. Finally, we have discussed the coupling ratio, showing that themotor loses the tight coupling as F increases. The efficiency is related with the coupling ratio andan analytical expression is given with a good agreement with the simulations.This work was supported by the Ministerio de Educacio´n y Ciencia (Spain) under the project No.BFM2003 − 07850 and the grant No. BES − 2004 − 3208(A.C.).[1] Visscher,K., Schnitzer,M.J., & Block,S.M. Nature 400, 184-189 (1999).[2] Hua,W. Chung,J & Gelles,J. Science 295 844-848 (2002).[3] Asbury,C.L.,Fehr,A.N. & Block,S.M. Science 302 2130-2134 (2003).[4] Stratopoulos,G.N., Dialynas,T.E. & Tsironis,G.P. Physics Letters A 252 151-156 (1999).[5] Dan,D., Jayannavar,A.M. & Menon,G.I. Physica A 318 40-47 (2003).[6] Fogedby,H.C., Metzler,R & Svane,A. PRE 70 021905 (2004).[7] Klumpp,S, Mielke,A & Wald,C. PRE 63 031914 (2001).[8] Mogilner,A., Mangel,M. & Baskin,R.J. Physics Letters A 297-306 (1998).[9] Downton,M.T., Zuckermann,M.J., Craig,E.M., Plischke,M & Linke,H. PRE 73 011909 (2006).[10] Ciudad,A. & Sancho,J.M. Biochemical Journal 390 345-349 (2005).

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Dynamics of an inchworm nano-walker

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