Solutions of burnt-bridge models for molecular motor transport

Transcription

1 PHYSICAL REVIEW E 75, Solutions of burnt-bridge models for moleular motor transort Alexander Yu. Morozov, Ekaterina Pronina, and Anatoly B. Kolomeisky Deartment of Chemistry, Rie University, Houston, Texas 77005, USA Maxim N. Artyomov Deartment of Chemistry, Massahusetts Institute of Tehnology, Cambridge, Massahusetts 0239, USA Reeived 23 Otober 2006; revised manusrit reeived 4 Deember 2006; ublished 2 Marh 2007 Transort of moleular motors, stimulated by interations with seifi links between onseutive binding sites alled bridges, is investigated theoretially by analyzing disrete-state stohasti burnt-bridge models. When an unbiased diffusing artile rosses the bridge, the link an be destroyed burned with a robability, reating a biased direted motion for the artile. It is shown that for robability of burning = the system an be maed into a one-dimensional single-artile hoing model along the eriodi infinite lattie that allows one to alulate exatly all dynami roerties. For the general ase of a theoretial method is develoed and dynami roerties are omuted exliitly. Disrete-time and ontinuoustime dynamis for eriodi distribution of bridges and different burning dynamis are analyzed and omared. Analytial reditions are suorted by extensive Monte Carlo omuter simulations. Theoretial results are alied for analysis of the exeriments on ollagenase motor roteins. DOI: 0.03/PhysRevE PACS numbers: 87.6.A I. INTRODUCTION Moleular motors, or motor roteins, lay an imortant role in the funtioning of biologial systems 3. They transform a hemial energy into mehanial work with a high effiieny and move along linear moleular traks. Conventional moleular motors fuel their motion by hydrolyzing adenosine trihoshate ATP or related omounds, and the hemial energy influx reates a biased direted motion along one-dimensional rotein filaments. During the transloation of motor roteins the hemial state of linear rotein traks remains unhanged. Reently, another tye of moleular motor, indeendent of ATP hydrolysis, has been disovered 4,5. A seial rotein alled ollagenase moves along ollagen fibrils, whih are the main omonent of extraellular matrix, and it leaves the filament at seial sites. Collagen roteolysis rovides a hemial energy that rodues a direted motion of the motor rotein moleule. It was found that the random motion of an enzyme moleule an be transformed into the biased diffusion along the linear trak if, after the leavage of ollagen, the motor rotein ould only move on one side of the broken link 5. It was suggested that the dynamis of ollagenases an be exlained by a so-alled burnt-bridge model BBM 4 7. Aording to this aroah, the motor rotein is viewed as a random walker hoing along the one-dimensional lattie that onsists of two tyes of links: Strong and weak. Strong links are not affeted when the artile asses them, however rossing the weak links might destroy them with a robability 0, and the random walker is not allowed to move again over the burnt links. Using a satial ontinuum aroximation that neglets the underlying lattie, the burntbridge model has been disussed in the limiting ases of very low and 6. This aroximation is only valid in the limit of very low onentrations of the seial links, although in biologial systems the density of bridges might be signifiant 4,5. Antal and Kraivsky 7 have investigated BBM by using a disrete-time random walk aroah that allowed one to alulate exatly veloities and disersions for some sets of arameters for several versions of the model. However, disrete-time dynamis is unrealisti for motor roteins sine their dynamis is tightly ouled to a set of hemial transitions 8, and time intervals between these transitions are distributed exonentially aording to Poisson statistis. A disrete-time statistis imliitly assumes a delta-funtion waiting time distribution between onseutive stes. In addition, the aroah resented in Ref. 7 is mathematially very involved, and not all relevant dynami roerties have been omuted. The goal of this aer is to investigate more realisti ontinuous-time disrete-state stohasti burnt-bridge models using simle analytial aroahes and extensive Monte Carlo omuter simulations. II. MODEL Let us onsider a single random walker that moves along an infinite one-dimensional lattie as shown in Fig.. The artile an ho in both diretions with the same rate v 0 =, i.e., initially there is no bias in the motion of the random walker. We set the size of the lattie saing to be equal to one. There are two tyes of links between onseutive binding sites on the lattie. The random walker does not interat with the strong links, while rossing the weak link leads to burning of the bridge with the robability. We assume that after the burning of the bridge the artile is always on the 0 2 N N FIG.. A shemati view of the motion of the artile in the system with burnt bridges. Thik solid lines are strong links, while thin solid lines are weak links, or bridges that an be burnt. Dotted lines are already burnt bridges. The artile an jum with equal rates to the right or to the left unless the link is broken /2007/753/ The Amerian Physial Soiety

2 MOROZOV et al. right side from it, and there are no burnt bridges at the initial time. Thus the artile is never traed between broken links, and it moves ontinuously forward. We distinguish two different ossibilities of burning the bridge. In the first otion, the bridge is burned only when the artile rosses it from left to right, but nothing haens when the walker moves from right to left. This model is alled a forward BBM. In the seond otion, the artile burns the bridge on any forward and bakward motion aross the weak link, desribing a forward-bakward BBM. It is imortant to note that both ases are idential for =, while for dynamis differs signifiantly for different burning senarios. Dynami roerties of the system also deend on the details of the artile behavior near the already burned bridge 7. In the standard version of BBM the artile sitting next to the broken link an only move forward in the right diretion. In the modified version of BBM the walker tries to ross the burnt bridge, but the attemt fails and the artile does not hange its osition. The dynamis of the motor rotein moleule in different BBMs is diretly determined by two arameters: The robability of burning and the onentration of weak links. The distribution of bridges strongly influenes the dynami roerties 7, and two different ases an be disussed: Periodi and random distributions. The ase when the bridges are found at equal distane N =/ lattie saings aart from eah other desribes the eriodi distribution. In this aer we will analyze in detail this situation, although our methods an be also alied to random distribution of bridges. III. BURNT-BRIDGE MODELS WITH = First, we onsider a seial ase of BBM with = when any rossing of the bridge leads to its burning. In addition, it is assumed that the first burnt bridge was rossed from the left. Thus the artile drifts in the right forward diretion. For the eriodi distribution of bridges the dynami roerties of the system an be evaluated exliitly by maing the model into the motion of the random walker on infinite onedimensional eriodi lattie, for whih exat solutions have been derived by Derrida in The artile at site j,,...,n moves forward bakward with the rate u j w j, and the eriodiity requires that u j u j+n and w j w j+n, where N is the size of the eriod. For this model the exliit exression for the mean veloity V is given by 9 V = N N r j N i= w i u i, while the disersion, or diffusion oeffiient, an be found from j D = u j s j r N r j N 2V N s j i=0 N i +r j+i+ + N VN +2, 2 2 with auxiliary funtions and V,D 0.2 PHYSICAL REVIEW E 75, V_sim D_sim V_theor D_theor 0 FIG. 2. Dynami roerties of a ontinuous-time burnt-bridge model with eriodi distribution of bridges for =. Lines are from analyti alulations, while symbols are obtained from Monte Carlo omuter simulations see text for details. r j = u j+ s j = u j+ N j+k i=j+ N j k i=j w i u i 3 w i+ u i. 4 The BBM with = orresonds to the artile hoing model with u j =w j = for all j with the exetion of w 0 =0, whih reflets the omlete burning of the bridge. Then from Eq. the exliit formula for the veloity is given by 2 V = N + = The disersion an be obtained from Eq. 2, yielding D = The dynami roerties of BBM with = are shown in Fig. 2 as funtions of the onentration of bridges. The veloity is a monotonially growing funtion of, while the disersion is always dereasing. It is interesting to onsider their behavior in limiting ases. When every link on the lattie is a bridge = we obtain V= and D=/2, as exeted. More interesting is the limit of 0. The veloity goes to zero, V2, as exeted, and this rerodues the result of the ontinuum theory of Mai et al. 6. But, surrisingly, it is found that D BBM =0=2/3 and it is not equal to the disersion of a free artile on the lattie without bridges, D free =. This jum in the disersion might orresond to a dynami transition, indiating qualitatively different behavior of the artile moving in the system without weak links and artile diffusing in BBM. Similar behavior has been observed in the disrete-time BBM 7. However, this interesting henomenon requires a more areful and detailed investigation. The maing of BBM to the artile hoing model on the eriodi lattie an also be used to investigate disrete

3 SOLUTIONS OF BURNT-BRIDGE MODELS FOR PHYSICAL REVIEW E 75, time burnt-bridge models. It was found 9 that in this ase the exliit exression for the veloity is the same as in the ontinuous ase, while for the diffusion oeffiient it an be shown that D disr =D ont V 2 /2, and D ont and V an be obtained from Eqs. and 2. However, the rate onstants u j and w j have to be reinterreted as juming robabilities, leading to ( ) ( ) u j = w j = for j =,2,...,N. 7 2 At the same time, for the artile near the burned bridge the rates are u 0 = and w 0 =0 for the standard BBM, while u 0 =/2 and w 0 =0 for the modified BBM. Then the dynami roerties of the standard disrete-time BBM are and for the modified BBM we obtain V =, D = 3 2, 8 V = +, D = These results fully agree with those obtained in Ref. 7 for eriodi bridge distribution. Note, however, that our aroah based on Derrida s analysis is muh more straightforward. IV. BURNT-BRIDGE MODELS WITH Derrida s method annot be easily extended to inlude the general ase of bridge burning with the robability sine it is not lear how the hoing rates u j and w j deend on. To alulate dynami roerties of BBM we introdue another method that is very general and aliable for many different systems. Below we analyze in detail several BBMs to illustrate our aroah. A. Continuous-time forward BBM for the eriodi distribution of bridges Now we onsider a ontinuous-time BBM with the artile destroying the link with the robability only when the bridge is rossed from left to right. The distribution of weak links is eriodi, i.e., they are ositioned N=/ sites aart from eah other. Let us define R j t as the robability to find the random walker j sites aart from the last burnt bridge at time t. The robabilities R j t are assoiated with the moving system of oordinates with the last burned bridge always at the origin. The orresonding kineti sheme is shown in Fig. 3a. The dynamis of the system an be desribed by a set of master equations. For the lattie sites not onneted to weak links or onneted only from the left side we have dr kn+i t = R kn+i t + R kn+i+ t 2R kn+i t, 0 dt for k=0,,2,... and i=,2,...,n. For all other lattie sites exet the origin one an show that dr kn t = R kn t + R kn+ t 2R kn t dt for k =,2,.... The dynamis are different at the origin see Figs. and 3a where dr 0 t dt = R kn t + R t R 0 t. In addition, there is a normalization ondition, N k=0 R kn+i t =, i=0 2 3 whih is valid at all times. In the stationary-state limit t we have d/dtr j t =0, and Eqs. 0 2 simlify signifiantly, and 0 2 N N N+ 2N (a) /2 0 2 ( )/2 N ( )/2 2N (b) R i = R 0 is, /2 /2 /2 ( )/2 S = R kn for i =0,,...,N 4 /2 R kn+i = i +R kn i R kn k =,2,.... ( )/2 FIG. 3. Redued kineti shemes for systems with eriodi distribution of bridges: a Continuous-time forward BBM only transition rates not equal to unity are shown; b disrete-time forwardbakward BBM only transition rates not equal to /2 are shown. The origin orresonds to the right end of the last burnt bridge. for i =0,,...,N, 5 To find the solutions of Eqs. 4 and 5 we assume that it has the following form: /

4 MOROZOV et al. R kn+i = R 0 e ak ibk, 6 where a and Bk are unknown arameters. This exression assumes a linear deendene between the weak links and an exonential derease for onseutive eriods. To eliminate Bk, we observe that Eqs. 4 and 5 simultaneously hold for i=n, giving us whih leads to R k+n = R 0 e ak+ = R kn+n = R 0 e ak NBk, 7 Bk = R 0 N eak e a. 8 To obtain the exression for a, we substitute R kn+i from Eq. 6 in the reurrent formula 5, R 0 e ak ibk = i +R 0 e ak i R 0 e ak N Bk. 9 Then using the exression 8 for Bk and defining xe a, Eq. 9 eventually redues to x 2 2+N x + =0. 20 Sine R kn+i is a dereasing funtion, the arameter a is always less than, and we hoose a hysially reasonable solution of Eq. 20, x =+ 2 N 2 2 N 2 +4N. 2 Note that generally 0x and x=0 for =, while x= for =0. The robability to find the artile next to the burnt bridge, R 0, an be found from the normalization ondition 3 and Eq. 2, yielding 2 x R 0 = N ++xn = 2 2 N 2 +4N 2 = , 22 and the solution for R kn+i in terms of x and R 0 is finally given as R kn+i = R 0 x k i N x. 23 Exat solutions for the robabilities R j to find the artile at given distane from the last burnt bridge allow us to alulate many dynami roerties of the system. Seifially, the mean veloity of the artile is given by V = u j w j R j. 24 However, sine the rates to go forward and bakward are the same at all sites exet the origin, all ontributions to the veloity anel out and only the term survives, 2 V, = R 0 = For = this equation redues to V, ==2/+, whih agrees exatly with the result 5 derived using the maing to Derrida s random hoing model. In the limit of 0 and 0 Eq. 25 beomes V as omared to V 2 in 6. The disreany is due to the fat that in our model only forward burning is onsidered, while in Ref. 6 the bridge an be burned on both forward and bakward motions of the motor rotein moleule. The exression for the veloity 25 also simlifies in the ase of N==, giving V=. Another useful dynami roerty that an be obtained from our analysis is average and average-squared distanes between the artile and the last burnt bridge, namely j = jr j, j = R 0 N2 j 2 = j 2 R j. 26 Substituting Eq. 23 into the exressions 26 we obtain x x 2 + N and j 2 = R 0 N3 xx + x NN2 x, x + 28 where x and R 0 are given by Eqs. 2 and 22. For = one an show that j = N /3, j 2 = NN /6. 29 In the limit of we have j N/, j 2 2N/, 30 while in the oosite limit of the results are jn/3, j 2 N 2 /6. 3 For the seial ase of N== we obtain j =, j 2 = We an also alulate the average and average-squared distanes between the onseutive burnt bridges. Beause the burning is a stohasti roess, i.e.,, this distane generally deviates from N. In order to omute these roerties we introdue P kn as a robability that the next bridge that will be destroyed is kn sites away from the given burnt bridge. Then P kn = PHYSICAL REVIEW E 75, R kn R kn, k =,2, The average and average-squared distanes between the onseutive burnt bridges an be found from the following exressions:

5 SOLUTIONS OF BURNT-BRIDGE MODELS FOR PHYSICAL REVIEW E 75, l = knp kn, l 2 = kn 2 P kn. 34 R kn+i t + = 2 R kn+i t + 2 R kn+i+t, 38 By substituting Eqs. 23 and 33 into 34 and using Eq. 2 we found that l = N x, l2 = N2 +x x The average distane l is related to the fration of unburnt bridges q via the following exression: l/n q = = N/l = x, l/n 36 whih gives the hysial meaning of the variable x used in our analysis. In the limiting ases we obtain the following results: l = N, l 2 = N 2, for =, l =/, l 2 =2/ /, for N = =, l = N/, l 2 =2N/, for, l = N, l 2 = N 2, for. 37 B. Disrete-time forward-bakward BBM for the eriodi distribution of bridges Our theoretial method is very general and it an be alied to any version of BBM. As another examle, we analyze a motion of the artile in disrete-time framework with eriodially loated bridges that an burn with the robability during the forward and bakward transitions. We again introdue a modified kineti sheme by utting the last burnt bridge at the origin, as shown in Fig. 3b. The imortant differene between ontinuous-time and disrete-time models is the fat that instead of transition rates in the ontinuous-time ase transition robabilities are used in the disrete-time dynamis. As a result, the sum of transition robabilities at eah site is equal to unity. We assume that at the sites that do not suort the weak links the transition robabilities are equal, i.e., u kn+i =w kn+i =/2 and w kn =u kn =/2 for any k and for i=,2,...,n 2. The transition robabilities are also equal at the sites that surround the weak links, giving u kn =w kn = /2 for,2,... see Fig. 3b. In addition, at the origin it is assumed that u 0 = and w 0 =0. As before, we define R j t as the robability to find the random walker at the osition j at time t. The dynamis of the system is governed by a set of disrete-time Master equations that are different from the ontinuous-time version 7,9. For the lattie sites that do not suort the weak links the orresonding master equations are for any k and with i=,2,...,n 2 with the exetion of R 0 and R see below. For the lattie sites around the bridges we have R kn t + = 2 R kn 2t + 2 R knt, R kn t + = 2 R kn t + 2 R kn+t, 39 for k =,2,... The dynamis is different near the origin, R 0 t + = 2 S + 2 R t, R t + = R 0 t + 2 R 2t, 40 where S= R kn t+r kn t. At large times the system reahes a stationary state with R kn+i t+=r kn+i t, and Eqs simlify to yield for the first eriod k=0, R i =2R 0 is for i =,2,...,N, R N = 2R 0 NS, 4 while for all other eriods k one an obtain R kn+i = i +R kn i R kn, for i =0,,...,N, R kn+n = N +R kn N R kn. 42 In order to solve Eqs. 4 and 42 it is natural to look for a solution in the form R kn+i =2R 0 e ak ibk, 43 for all k and i exet the origin k=i=0. Note that the fator of 2 in front of R 0 in Eq. 43 is neessary to ensure the onsisteny between the solutions for k=0 from Eq. 4 and for k see Eq. 42. The solutions an be obtained using the same aroah as was disussed in the revious setion for the ontinuous-time forward BBM. The final exressions are R 0 = x N + N N x, R kn+i =2R 0 x k i N x, 44 where we again define xe a, and it an be shown that x = 2 2 4, = N + N. 45 The dynami roerties of the artile in this model an be alulated as resented above for the ontinuous-time for

6 MOROZOV et al. PHYSICAL REVIEW E 75, ward BBM. For the veloity we obtain the following formal exression: v = R 0 + R kn The validity of this equation an be seen from the fat that the forward and bakward hoing rates anel eah other on all sites exet at the origin and the sites labeled kn k =,2,... From Eq. 44 it follows that R kn =2R 0 x k. Substituting this result into Eq. 46 yields a b V 0.2 d + x v = R 0, 47 x that an be transformed with the hel of Eq. 44 into (a) 0 x v = N + N N x. 48 Finally, relaing x aording to Eq. 45 and using N=/ we obtain where v, = 2 + Z 2 Z, Z = After some rearrangement, it an be shown that this formula is idential to the exression for the veloity obtained by Antal and Kraivsky 7. But note again that our derivation is muh simler. V. DISCUSSIONS To hek the validity of our theoretial aroah we erformed extensive Monte Carlo omuter simulations for several BBMs. A general algorithm for ontinuous-time and disrete-time random walks, disussed in detail in Ref., has been utilized. Veloities have been alulated from the single simulations, onsisting of stes, via the basi formula V=x/t. Diffusion onstants have been determined using the exression D=x 2 x 2 /2t, and tyially more than 0 5 simulation runs have been averaged over to redue the stohasti noise in the data. The results of simulations, as shown in Figs. 2, 4, and 6, in all ases are in exellent agreement with available analytial alulations, suorting our analytial reditions. Theoretial alulations show that the veloity of the artile in BBM is a monotonially inreasing funtion of the onentration of weak links and burning robability: see Figs. 2, 4, and 5. However, the behavior of the disersion is more omlex, as an be judged from Figs. 2 and 6. For fixed values of it dereases as a funtion of, although note large flutuations when 0. Inrease in the burning (b) V FIG. 4. a Veloity as a funtion of onentration of bridges for different burning robabilities for the eriodi forward ontinuoustime BBM. Solid lines are results of analytial alulations, and symbols are from Monte Carlo simulations. Curve a orresonds to =0.9, urve b orresonds to =0.7, urve orresonds to =0.5, and urve d orresonds to =0.. b Veloity as a funtion of burning robability for different onentrations of weak links for the eriodi forward ontinuous-time BBM. Solid lines are results of analytial alulations and symbols are from Monte Carlo simulations. Curve a orresonds to =, urve b orresonds to =/2, urve orresonds to =/3, and urve d orresonds to =/0. robability first lowers signifiantly the disersion, and then it starts to inrease slowly. Our omuter simulations indiate see Fig. 6a that when every link in the system is a otential bridge = the disersion is always equal to /2 for any 0. Our theoretial omutations also indiate that details of burning roess forward versus forward-bakward are affeting the dynamis of the artile. Under the same onditions, the random walker moves faster in the forwardbakward BBM, and this effet is signifiant for large and : see Fig. 5. These results are exeted sine the burning of the bridge fuels the forward motion of the artile. The ossibility of burning in the forward and bakward motion aelerates the motion of the random walker. Develoed theoretial method allows us to analyze exerimental data on the motion of ollagenases motor roteins 4. Using fluoresene orrelation setrosoy it was determined that the diffusion oeffiient is D =8±.5 a b d

7 SOLUTIONS OF BURNT-BRIDGE MODELS FOR V FIG. 5. Veloity as a funtion of onentration for different burning realizations for the ontinuous-time eriodi BBM. Solid lines are results of analytial alulations. Curve a orresonds to forward-bakward BBM with =, urve b orresonds to forward BBM with =, urve orresonds to forward-bakward BBM with =0.2, and urve d orresonds to forward BBM with = m 2 s and the mean drift veloity is V=4.5± ms 4. Our theoretial results are obtained assuming that the lattie saing and transition rates are equal to unity. To restore the orret units we assume that the motor rotein moves in disrete stes of size a. Although the exat D (a) 0.7 D 0.5 a b =0. =0.5 = =0. =0.5 = (b) FIG. 6. Disersion in the forward ontinuous-time eriodi BBM from Monte Carlo simulations a as a funtion of burning robability and b as a funtion of onentration of bridges. d PHYSICAL REVIEW E 75, value of a is unknown we an reasonably take a0 nm, whih is of the same order as ste sizes of other moleular motors 2. From the struture of the ollagen it is known that leavage sites, that orresond to the bridges, are searated by the distane =300 nm,4. It suggests that BBM with eriodi distribution of bridges and =a/=/30 is aroriate to desribe the dynamis of the system. Sine at low onentrations the reditions for the dynami roerties in the forward and forward-bakward BBM are very similar see Fig. 5, for the veloity we use Eq. 25 multilied by av 0, 2 V, = av , 5 where v 0 is the transition rate. For the diffusion oeffiient analytial results are not known, but for the limiting result of =0 an be used, yielding D = a 2 v Equations 5 and 52 with two unknowns, and v 0, an be easily solved, and we obtain 0.28, v s. 53 Our estimate of the burning robability 28% is larger than the result =0% obtained in Monte Carlo simulations 4. However, the differene an be attributed to different versions of BBM used in our theoretial analysis and in omuter simulations 4, and to the unertainty in the determination of the ste size a. It is also interesting to analyze the effiieny of the motor rotein driven by burning of bridges. Our theoretial analysis indiates that the motion of the artile in the system with burnt bridges an be viewed as a biased random walk. We assume that the dynami roerties of the random walker in BBM an be effetively written as V = au eff w eff, D = 2 a2 u eff + w eff, 54 where u eff and w eff are effetive transition rates of the biased random walk. Then the fore exerted by the moleular motor an be found from 0,2 F = k BT a ln u eff w eff. 55 Substituting exerimental values for V and D into Eqs. 54 to obtain the effetive rates u eff and w eff, the exerted fore an be alulated, yielding F0.023 N, whih is two orders of magnitude weaker than the fores rodued by other motor roteins 2. However, this fore is exerted over the distane of =300 nm, whih is 0 00 times larger than for onventional moleular motors, roduing W=F 6.9 Nnm of work. The leavage of three etide bonds simultaneously, that fuels the motion of ollagenase, liberates E62.7 kj/mole or 04.4 Nnm of energy 4. The alulated effiieny of the ollagenase motor rotein, =W/E6.3%, is only slightly lower than the effiieny of 5% obtained from the exerimental estimates 4. Thus single ollagenase rotein is a weak moleular motor and its

8 MOROZOV et al. effiieny is not very high in omarison with other motor roteins. However, it was suggested reently 5 that ouling of several enzymes might rodue a very effiient motor rotein. VI. SUMMARY AND CONCLUSIONS A general theoretial analysis of artile dynamis in different one-dimensional burnt-bridge models is resented. It is shown that, when the robability of burning is equal to unity, the system an be maed into the model of the single artile hoing along the eriodi infinite lattie. This allows us to obtain all dynami roerties in exat and exliit form. For the general ase of we develoed a method of analyzing the dynamis by onsidering a redued kineti sheme with reset to the last burnt bridge. It is found that distribution of weak links and the details of the burning mehanism strongly influene the artile dynamis in BBM. Theoretial alulations are fully suorted by Monte Carlo omuter simulations. Aliation of this theoretial method to exeriments on ollagenase motor roteins rovides the estimates of the burning robability and the exerted fore, leading to the onlusion that this moleular motor is rather weak and not very effiient, in ontrast to onventional motor roteins. PHYSICAL REVIEW E 75, Although the resented theoretial aroah investigates exliitly the dynami roerties of many BBMs and it is suessfully alied for analysis of exerimental data, there are several roblems that should be addressed in the future studies. We have not been able to obtain exressions for the diffusion oeffiient in the general ase of, and our analysis is given only for the eriodi distribution of weak links. It is also not lear what is the origin of jum in the disersion in the limit of 0. It will be interesting to investigate the motion of the motor rotein on latties with eriodi rates 0 and with bridge burning, and it is reasonable to suggest that the aroah develoed in this aer an be generalized for this ase too. In addition, a more realisti aroah will be to allow the moleular motor to jum between arallel traks 4. Future exeriments on single motor roteins that utilize bridge burning mehanisms will be imortant to test the validity of the resented theoretial aroah. ACKNOWLEDGMENTS The authors would like to aknowledge the suort from the Welh Foundation Grant No. C-559, the U.S. National Siene Foundation Grant No. CHE , and the Hammill Innovation Award. H. Lodish et al., Moleular Cell Biology W. H. Freeman and Comany, New York, J. Howard, Mehanis of Motor Proteins and Cytoskeleton Sinauer Assoiates, Sunderland, MA, D. Bray, Cell Movements. From Moleules to Motility Garland Publishing, New York, S. Saffarian, I. E. Collier, B. L. Marmer, E. L. Elson, and G. Goldberg, Siene 306, S. Saffarian, H. Qian, I. E. Collier, E. L. Elson, and G. Goldberg, Phys. Rev. E 73, J. Mai, I. M. Sokolov, and A. Blumen, Phys. Rev. E 64, T. Antal and P. L. Kraivsky, Phys. Rev. E 72, A. B. Kolomeisky and M. E. Fisher, J. Chem. Phys. 3, B. Derrida, J. Stat. Phys. 3, A. B. Kolomeisky and M. E. Fisher, Annu. Rev. Phys. Chem. 58, J. F. MCarthy, J. Phys. A 26, M. E. Fisher and A. B. Kolomeisky, Pro. Natl. Aad. Si. U.S.A. 96,