Dynamics of molecular motors in reversible burnt-bridge models
Dynamic properties of molecular motors whose motion is powered by interactions with specific lattice bonds are studied theoretically with the help of discrete-state stochastic "burnt-bridge" models. Molecular motors are depicted as random walkers that can destroy or rebuild periodically di...
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| Zitieren: | Dynamics of molecular motors in reversible burnt-bridge models / M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23801: 1-18. — Бібліогр.: 10 назв. — англ. |
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| author | Artyomov, M.N. Morozov, A.Yu. Kolomeisky, A.B. |
| author_facet | Artyomov, M.N. Morozov, A.Yu. Kolomeisky, A.B. |
| citation_txt | Dynamics of molecular motors in reversible burnt-bridge models / M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23801: 1-18. — Бібліогр.: 10 назв. — англ. |
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| description | Dynamic properties of molecular motors whose motion is powered by interactions with specific lattice bonds are studied theoretically with the help of discrete-state stochastic "burnt-bridge" models. Molecular motors are depicted as random walkers that can destroy or rebuild periodically distributed weak connections ("bridges") when crossing them, with probabilities p1 and p2 correspondingly. Dynamic properties, such as velocities and dispersions, are obtained in exact and explicit form for arbitrary values of parameters p1 and p2. For the unbiased random walker, reversible burning of the bridges results in a biased directed motion with a dynamic transition observed at very small concentrations of bridges. In the case of backward biased molecular motor its backward velocity is reduced and a reversal of the direction of motion is observed for some range of parameters. It is also found that the dispersion demonstrates a complex, non-monotonic behavior with large fluctuations for some set of parameters. Complex dynamics of the system is discussed by analyzing the behavior of the molecular motors near burned bridges.
|
| first_indexed | 2025-12-07T17:51:12Z |
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Condensed Matter Physics 2010, Vol. 13, No 2, 23801: 1–18
http://www.icmp.lviv.ua/journal
Dynamics of molecular motors in reversible
burnt-bridge models
M.N. Artyomov1, A.Yu. Morozov2, A.B. Kolomeisky3
1 Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
2 Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles, CA 90095, USA
3 Department of Chemistry, Rice University, Houston, TX 77005, USA
Received November 23, 2009
Dynamic properties of molecular motors whose motion is powered by interactions with specific lattice bonds
are studied theoretically with the help of discrete-state stochastic “burnt-bridge” models. Molecular motors are
depicted as random walkers that can destroy or rebuild periodically distributed weak connections (“bridges”)
when crossing them, with probabilities p1 and p2 correspondingly. Dynamic properties, such as velocities
and dispersions, are obtained in exact and explicit form for arbitrary values of parameters p1 and p2. For the
unbiased random walker, reversible burning of the bridges results in a biased directed motion with a dynamic
transition observed at very small concentrations of bridges. In the case of backward biased molecular motor
its backward velocity is reduced and a reversal of the direction of motion is observed for some range of
parameters. It is also found that the dispersion demonstrates a complex, non-monotonic behavior with large
fluctuations for some set of parameters. Complex dynamics of the system is discussed by analyzing the
behavior of the molecular motors near burned bridges.
Key words: molecular motors, stochastic models, motor proteins
PACS: 87.16.Ac
1. Introduction
In recent years an increased attention has been devoted to investigations of molecular motors,
also known as motor proteins, that are crucial in many cellular processes [1]. They transform
chemical energy into the mechanical motion in non-equilibrium conditions. For most of molecular
motors their motion along linear molecular tracks is fueled by the hydrolysis of adenosine tri-
phosphate (ATP) or related compounds. It was suggested that a different mechanism is employed
to power the motion of a protein collagenase along collagen fibrils [2, 3]. It probably utilizes the
collagen proteolysis, cleaving the filament at specific sites. As the collagenase molecule is unable
to cross the already broken bond, it leads to the biased diffusion along the filament. However, full
understanding of mechanisms of collagenase motion is still not available.
It was proposed that a good description of the collagenase dynamics could be provided by
the so-called “burnt-bridge model” (BBM) [2–9]. In this model, the motor protein is depicted as
a random walker that translocates along the one-dimensional lattice that consists of strong and
weak bonds. While the strong bonds remain unaffected if crossed by the walker in any direction,
the weak ones (termed “bridges”) might be broken (or “burnt”) with a probability 0 < p1 6 1
when crossed in the specific direction, and the walker cannot cross the burnt bridges again, unless
they are restored, which can occur with probability 0 < p2 6 1. In [6, 7] an analytical approach
was developed which permitted us to derive the explicit formulas for molecular motor velocity
V (c, p1) and diffusion constant D(c, p1) for the entire ranges of burning probability 0 < p1 6 1
and concentration of the bridges 0 < c 6 1 which were also confirmed by extensive Monte Carlo
computer simulations. This theoretical method has been applied to several problems with periodic
bridge distribution. However, the results in [6, 7] have been obtained only for irreversible bridge
burning (bridge recovery probability was taken to be p2 = 0), and also for unbiased random walker
between bridges (equal forward and backward transition rates). In present work, we generalize
c© M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky 23801-1
http://www.icmp.lviv.ua/journal
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
our approach to allow for the possibility of bridge recovery as well as unequal hopping rates on
the sites between bridges. It is more realistic to consider systems with reversible action of motor
proteins since they are catalysts that equally accelerate both forward and backward biochemical
transitions [1].
2. Model
According to our model, we view a motor protein as a random walker moving along an infinite
one-dimensional lattice with forward and backward transition rates being u and w correspondingly,
as illustrated in figure 1. The lattice spacing size is set to be equal to one. The lattice is composed
of strong and weak bonds. There is no interaction between the random walker and strong bonds.
However, crossing the bridge in the forward direction (from left to right) leads to its burning with
the probability p1, while the particle moves with the rate u. After the weak link is destroyed, the
walker is assumed to be on the right side of it. When the particle is trying to cross a broken bond,
the bridge can be recovered with the probability p2, while the particle moves to the left with rate
w. It is assumed that initially, at t = 0, all bridges are intact.
p1
p2
w u
−1 0 1 2 N−1 N
Figure 1. A schematic picture of the motion of a molecular motor in the reversible burnt-
bridge model. Thick solid lines depict strong links, while thin solid lines represent periodically
distributed weak links (bridges). Dotted lines are for already burnt bridges.
The details of breaking weak bonds in BBM have a strong effect on the dynamic properties
of motor proteins [6]. There are two different possibilities of bridge burning. In the first variant
(the so-called “forward BBM”), the weak bond is broken when crossed from left to right, but the
intact bridge is not affected when the particle moves from right to left. Thus the bridge recovery
may occur if the walker attempts to cross a burnt bridge from right to left. In the second variant
(named “forward-backward BBM”), the weak link is destroyed if crossed in either direction [4, 5].
Both variants are identical for p1 = 1, however for p1 < 1 the dynamics is different in two burning
scenarios, as was shown in p2 = 0 case [6], although mechanisms are still the same. For reasons
of simplicity, below we will only consider forward BBM, even though forward-backward BBM can
also be solved using the same method.
There are five parameters that specify the dynamics of molecular motors in BBM: the prob-
abilities p1, p2, the concentration of bridges c, as well as transition rates u and w. The dynamic
properties of the walker are also strongly effected by the distribution of weak bonds [5]. Below we
will study the case of periodically distributed bridges, when their concentration is c = 1/N and the
weak bonds are located between the lattice sites with the coordinates kN −1 and kN , with integer
k (see figure 1). This description is more realistic for collagenases’ dynamics [2, 3]. The model
below will be studied using continuous time analysis as it better describes chemical transitions in
motor proteins [6].
3. Dynamic properties for BBM with bridge recovery
3.1. Velocity
To find the walker’s velocity, we generalize the method used in [6] (for irreversible bridge
burning, i.e. p2 = 0) to allow for non-zero probability p2. We introduce a probability Rj(t) that
the random walker is found j sites apart from the last burnt bridge at time t. The probabilities
Rj(t) arise if the system is viewed in moving coordinate frame with the last burnt bridge always
at the origin, as illustrated by a reduced chemical kinetic scheme shown in figure 2.
23801-2
Reversible burnt-bridge models
1
p2 f(1)
2
2
p
p
p
0 1 2 N
N−1
1−p 1−p
2N
p f(2)
p f(3)
1 1
1
1
Figure 2. Reduced kinetic scheme for continuous-time forward BBM with bridge recovery (only
transition rates not equal to u, w are shown). The origin is the right end of the last burnt bridge.
Parameters are described in detail in the text.
The dynamics of the system is determined by a set of master equations:
dRkN+i(t)
dt
= uRkN+i−1(t) + wRkN+i+1(t) − (u + w)RkN+i(t), (1)
for k = 0, 1, 2, · · · and i = 1, 2, · · · , N − 2; and
dRkN+N−1(t)
dt
= uRkN+N−2(t) + wp2f(k + 1)R0(t) + wR(k+1)N (t) − (u + w)RkN+N−1(t), (2)
dR(k+1)N (t)
dt
= (1 − p1)uRkN+N−1(t) + wR(k+1)N+1(t) − (u + w)R(k+1)N (t), (3)
with k = 0, 1, 2, · · · for both equations (2) and (3). Also at the origin we have
dR0(t)
dt
= p1u
∞
∑
k=1
[RkN−1(t)] + wR1(t) − uR0(t) − wp2R0(t). (4)
In equation (2) we introduced a function f(k) as a probability that next to the last burnt bridge
is k periods to the left from the last burnt bridge. It satisfies the condition
∞
∑
k=1
f(k) = 1, which is
reflected in equation (4). The system of equations (1)–(4) is to be solved in the stationary-state
limit (at large times) when dRj(t)/dt = 0 is satisfied, and we denote Rj(t → ∞) ≡ Rj in what
follows. By definition, it can be argued that
f(k) =
p1RkN−1
∞
∑
k=1
p1RkN−1
. (5)
The solution of the system (1)–(4) can be facilitated by rewriting equation (5) in a more convenient
form. To this end, we note that based on the results from [6, 8] it is reasonable to assume that
RkN+i is of the form
RkN+i = ykW (i), (6)
where y and W are some functions of p1, p2, u, w and N . Furthermore, y is i and k-independent,
while W depends on i (but not on k). The advantage of the ansatz (6) is that it leads to a simpler
form of f(k):
f(k) = yk−1(1 − y), (7)
as follows from equation (5). We proceed to solve equations (1)–(4) with f(k) in equation (2) given
by the expression (7).
23801-3
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
Introducing a parameter β ≡ u/w, it can be easily verified that
RkN+i = C1(k) + C2(k)βi (8)
with arbitrary C1(k), C2(k) solves equation (1). Utilizing equation (8), C1(k), C2(k) can be ex-
pressed in terms of functions RkN and RkN+1 (first two points of the period). Then equation (8)
leads to
RkN+i = RkN − w
u − w
[
1 − βi
]
∆k , (9)
where we defined
∆k = RkN+1 − RkN . (10)
In expression (9) it was assumed that k = 0, 1, 2, · · · and i = 0, 1, · · · , N − 1. Parameters RkN
and ∆k are to be determined from equations (2) and (3). Substituting equation (9) in (2) and (3)
yields
uRkN − uw
u − w
[
1 − βN−2
]
∆k + wp2y
k(1 − y)R0 + wR(k+1)N − (u + w)RkN
+ (u + w)
w
u − w
[
1 − βN−1
]
∆k = 0, (11)
(1 − p1)uRkN − (1 − p1)
uw
u − w
[
1− βN−1
]
∆k + wR(k+1)N
− w2
u − w
[1 − β] ∆k+1 − (u + w)R(k+1)N = 0. (12)
Equation (11) also yields
∆k =
u − w
w(1 − βN )
[
RkN − R(k+1)N − p2y
k(1 − y)R0
]
. (13)
Substituting (13) into equation (12) results in
aRkN + bR(k+1)N + cR(k+2)N + (d + ey + fy2)R0y
k = 0, (14)
where
a = (1 − p1)u
[
1 − 1 − βN−1
1 − βN
]
, (15)
b = −u + (1 − p1)u
1 − βN−1
1 − βN
− w
1 − β
1 − βN
, (16)
c = w
1 − β
1 − βN
, (17)
d = p2(1 − p1)u
1 − βN−1
1 − βN
, (18)
e = −p2(1 − p1)u
1 − βN−1
1 − βN
+ wp2
1 − β
1 − βN
, (19)
and
f = −wp2
1− β
1 − βN
. (20)
Using equation (6), it follows that
RkN = R0y
k, (21)
and equation (14) turns into
(c + f)y2 + (b + e)y + (a + d) = 0. (22)
23801-4
Reversible burnt-bridge models
Therefore,
y =
−(b + e) −
√
(b + e)2 − 4(a + d)(c + f)
2(c + f)
, (23)
where we selected the solution of (22) such that 0 6 y < 1. With the help of (21) we obtain from
equation (13)
∆k =
u − w
w {1 − βN}R0y
k(1 − y)(1 − p2). (24)
Thus (9) yields
RkN+i = R0y
k
[
1 − (1 − p2)(1 − y)
1 − βi
1 − βN
]
. (25)
Equation (25) with y given by (23) solves the system of equations (1)–(4) in the stationary state
limit. We note that although the equation (4) was not used to find this solution, it was numerically
verified that every equation in the system (1)–(4) is indeed solved by expressions (25) and (23).
Parameter R0, needed to find the velocity, is found from equation (25) combined with the
normalization condition
∞
∑
k=0
N−1
∑
i=0
RkN+i = 1, which produces
R0 = (1 − y)
{
N − (1 − p2)(1 − y)
[
N
1 − βN
− 1
1 − β
]}−1
. (26)
The mean velocity of the walker is given by [6]
V =
∞
∑
j=0
(uj − wj)Rj = (u − wp2)R0 +
[
∞
∑
k=0
N−1
∑
i=0
(u − w)RkN+i
]
− (u − w)R0 , (27)
which results in a simple relation,
V = w(1 − p2)R0 + (u − w). (28)
In equation (28), R0 is given by (26) with y from the expression (23).
It can be shown that in the limit of u → 1, w → 1, and p2 → 0 equation (28) reproduces
the result obtained earlier in [6] for the BBM with u = w = 1 and p2 = 0. Also, equation (28)
considerably simplifies in the limiting case of p1 = 1 (deterministic bridge burning) when y = 0.
In the case of p1 = 1 we obtained V (u, w, p2, N) in [9] using the Derrida’s method [10] and our
general result given in equation (28) agrees with it in the p1 → 1 limit, as was numerically verified.
3.2. Diffusion coefficient
The diffusion coefficient is found by generalizing the method developed in [7] (where we found
dynamic properties of the random walker in BBM with u = w = 1, p2 = 0 and periodic bridge
distribution), allowing for 0 < p2 6 1 and u 6= w. We define PkN+i,m(t) as the probability that at
time t the random walker is located at point x = kN + i (i = 0, 1, · · · , N − 1), the right end of the
last burnt bridge being at the point mN . Parameters m and k > 0 assume integer values.
The dynamics of the system is described by a set of Master equations:
dPmN,m(t)
dt
= wPmN+1,m(t) − (u + p2w)PmN,m(t) + p1u
m−1
∑
m′=−∞
PmN−1,m′(t), (29)
for k = i = 0,
dPmN+kN+(N−1),m(t)
dt
= wPmN+(k+1)N,m(t) + uPmN+kN+(N−2),m(t)
+ p2wf(k + 1)PmN+(k+1)N,m+k+1(t) − (u + w)PmN+kN+(N−1),m(t), (30)
23801-5
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
for k > 0 and i = N − 1 [with f(k + 1) the same as in (2)],
dPmN+kN,m(t)
dt
= wPmN+kN+1,m(t) + (1 − p1)uPmN+kN−1,m(t) − (u + w)PmN+kN,m(t) (31)
for k > 1 and i = 0, and
dPmN+kN+i,m(t)
dt
= wPmN+kN+i+1,m(t) + uPmN+kN+i−1,m(t) − (u + w)PmN+kN+i,m(t) (32)
for k > 0 and i = 1, · · · , N − 2.
We observe that
RkN+i(t) =
+∞
∑
m=−∞
P(m+k)N+i,m(t), (k > 0; i = 0, 1, · · · , N − 1), (33)
where RkN+i(t) is the probability for the random walker to be found kN + i sites apart from the
last burnt bridge at time t, which was used above to find the walker’s velocity and is given by
equation (25). Plugging (33) into equations (29)–(32) results in the equations (1)–(4) for RkN+i,
thus obtained by a different method.
In accordance with [7, 10], we introduce auxiliary functions SkN+i(t),
SkN+i(t) =
+∞
∑
m=−∞
(mN + kN + i)PmN+kN+i,m(t), (k > 0 i = 0, 1, · · · , N − 1). (34)
The system of equations describing the time evolution of the functions SkN+i(t) results from
equations (34) and (29)–(32). It was obtained that
dS0(t)
dt
= wS1(t) − (u + p2w)S0(t) + p1u
∞
∑
α=1
SαN−1(t) + p1u
∞
∑
α=1
RαN−1(t) − wR1(t), (35)
dSkN+(N−1)(t)
dt
= wS(k+1)N (t) + uSkN+(N−2)(t) + p2wf(k + 1)[S0 − R0]
− (u + w)SkN+(N−1)(t) − wR(k+1)N (t) + uRkN+(N−2)(t), (36)
for k > 0,
dSkN (t)
dt
= wSkN+1(t)+(1−p1)uSkN−1(t)−(u+w)SkN (t)+(1−p1)uRkN−1(t)−wRkN+1(t) (37)
for k > 1 and
dSkN+i(t)
dt
= wSkN+i+1(t) +uSkN+i−1(t)− (u + w)SkN+i(t) +uRkN+i−1(t)−wRkN+i+1(t) (38)
for k > 0 and i = 1, · · · , N − 2.
At t → ∞ the solutions of equations (35)–(38) are sought in the form
Sj(t) = ajt + Tj , (39)
where aj and Tj are time-independent coefficients. Plugging equation (39) into equations (35)–(38)
leads to,
wa1 − (u + p2w)a0 + p1u
∞
∑
α=1
aαN−1 = 0, (40)
wa(k+1)N + uakN+(N−2) + p2wf(k + 1)a0 − (u + w)akN+(N−1) = 0 (41)
23801-6
Reversible burnt-bridge models
for k > 0,
wakN+1 + (1 − p1)uakN−1 − (u + w)akN = 0 (42)
for k > 1 and
wakN+i+1 + uakN+i−1 − (u + w)akN+i = 0 (43)
for k > 0 and i = 1, · · · , N − 2.
Clearly, equations (40)–(43) are identical to the system of equations (1)–(4) for the functions Rj
in the t → ∞ limit, where dRj/dt = 0. Thus their solutions should coincide up to the multiplicative
constant, namely,
akN+i = CRkN+i , (44)
with RkN+i given by equation (25). The normalization condition
∞
∑
k=0
N−1
∑
i=0
RkN+i = 1 implies that
C =
∞
∑
k=0
N−1
∑
i=0
akN+i. To find the explicit expression for C, we utilize the equations for Tj obtained
by plugging equation (39) into equations (35)–(38),
a0 = wT1 − (u + p2w)T0 + p1u
∞
∑
α=1
TαN−1 + p1u
∞
∑
α=1
RαN−1 − wR1 , (45)
akN+(N−1) = wT(k+1)N + uTkN+(N−2) + p2wf(k + 1)[T0 − R0] − (u + w)TkN+(N−1)
+ uRkN+(N−2) − wR(k+1)N (46)
for k > 0,
akN = wTkN+1 + (1 − p1)uTkN−1 − (u + w)TkN + (1 − p1)uRkN−1 − wRkN+1, (47)
for k > 1 and
akN+i = wTkN+i+1 + uTkN+i−1 − (u + w)TkN+i + uRkN+i−1 − wRkN+i+1, (48)
for k > 0 and i = 1, · · · , N −2. Summing up equations (45)–(48) and using
∞
∑
k=0
f(k +1) = 1 yields
C =
∞
∑
k=0
N−1
∑
i=0
akN+i = w(1 − p2)R0 + (u − w) = V, (49)
where the walker’s velocity V is given by (28). Hence, in accordance with equation (44)
akN+i = V RkN+i (50)
where RkN+i and V are given by equations (25) and (28).
Now we are able to obtain the expression for the random walker’s velocity [7]. The mean position
of the particle is given by
〈x(t)〉 =
+∞
∑
m=−∞
∞
∑
k=0
N−1
∑
i=0
(mN + kN + i)PmN+kN+i,m(t)
=
∞
∑
k=0
N−1
∑
i=0
{
+∞
∑
m=−∞
(mN + kN + i)PmN+kN+i,m(t)
}
=
∞
∑
k=0
N−1
∑
i=0
SkN+i(t), (51)
which results in the mean velocity
Ṽ =
d
dt
〈x(t)〉 =
∞
∑
k=0
N−1
∑
i=0
d
dt
SkN+i(t). (52)
23801-7
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
In the t → ∞ limit equations (39) and (49) therefore yield
Ṽ =
∞
∑
k=0
N−1
∑
i=0
akN+i = w(1 − p2)R0 + (u − w) = V. (53)
As expected, equation (53) reproduces the expression (28) for V obtained above with the use of
the reduced chemical kinetic scheme method.
For the purposes of computing the diffusion coefficient D, functions TkN+i need to be found
from (45)–(48) [7, 10]. In (45)–(48), akN+i should be expressed according to (50). In analogy with
finding RkN+i from (1)–(4), we start with solving (48). In the expression (25) for RkN+i, we denote
ξ =
(1 − p2)(1 − y)
1− βN
, (54)
so that
RkN+i = RkN [1 − ξ(1 − βi)], (55)
where RkN = R0y
k, with R0, y given by (26), (23). With the use of (55), (48) takes the form
wTkN+i+1 + uTkN+i−1 − (u + w)TkN+i + (u−w− V )RkN (1− ξ) + (w −u−V )RkN ξβi = 0. (56)
We seek the solution of (56) in the form
TkN+i = C1(k) + C2(k)βi + iA(k) + iB(k)βi. (57)
In (57), C1(k) + C2(k)βi part is the solution of homogeneous equation [the part of (56) which
involves only TkN+i, TkN+i±1], in analogy with equation (8). Substituting equation (57) into equa-
tion (56) gives these expressions for A(k) and B(k):
A(k) =
u − w − V
u − w
RkN (1 − ξ), B(k) =
u − w + V
u − w
RkNξ. (58)
Plugging TkN , TkN+1 instead of TkN+i into (57), one can express C1(k) and C2(k) in terms of
A(k), B(k) as well as TkN and TkN+1 (first two points of the period). Substituting the resulting
expressions for C1(k) and C2(k) together with the expressions (58) for A(k), B(k) in (57) gives
after some rearrangement
TkN+i = TkN + θk
1 − βi
1 − β
+ RkN
u − w − V
u − w
(1 − ξ)
[
i − 1 − βi
1 − β
]
+ RkN
u − w + V
u − w
ξ
[
iβi − β
1 − βi
1 − β
]
, (59)
where
θk = TkN+1 − TkN . (60)
Equation (59) solves (56) [i.e., the equation (48)] for arbitrary TkN and θk. Parameters TkN and
θk are to be determined from the equations (46) and (47). To get a less cumbersome result for
TkN+i, it is convenient to rewrite equation (46) by adding and subtracting wTkN+N , wRkN+N
[we note that TkN+N , RkN+N are given by (59) and (55) with i = N , i.e., TkN+N 6= T(k+1)N ,
RkN+N 6= R(k+1)N ]:
akN+(N−1) = wT(k+1)N − wTkN+N + wTkN+N + uTkN+(N−2) + p2wf(k + 1)[T0 − R0]
− (u + w)TkN+(N−1) + uRkN+(N−2) − wRkN+N + w(RkN+N − R(k+1)N ), (61)
which gives
w(T(k+1)N − TkN+N ) + w(RkN+N − R(k+1)N ) + p2wf(k + 1)[T0 − R0] = 0, (62)
23801-8
Reversible burnt-bridge models
as follows from (48) which was formally extended to include i = N − 1. In (62), the function
f(k + 1) is expressed according to equation (7). Plugging (59), (55) (with appropriate k, i) into
(62) permits us to express θk in terms of TkN in this way,
θk =
1 − β
1 − βN
{
T(k+1)N − TkN − FRkN + p2y
k(1 − y)T0
}
, (63)
where
F =
u − w − V
u − w
(1 − ξ)
[
N − 1 − βN
1 − β
]
+
u − w + V
u − w
ξ
[
NβN − β
1 − βN
1− β
]
. (64)
It should be mentioned that substituting (59) and (55) into original equation (46) leads to a more
involved expression connecting θk and TkN which turns out to be identical to equation (63), as was
numerically verified.
To find TkN , we substitute (59) and (55) (with appropriate k, i) in equation (47), with θk in
(59) replaced according to (63). After some algebra, this results in
aTkN + bT(k+1)N + cT(k+2)N + {d + ey + fy2}T0y
k + χyk = 0, (65)
where coefficients a, b, c, d, e, f are given by equations (15)–(20) and
χ = −R0F
[
wy
1 − β
1 − βN
+ (1 − p1)u
1 − βN−1
1 − βN
]
+ R0
{
(1 − p1)uφ − wy[1 − ξ(1 − β)]
+(1− p1)u[1 − ξ(1 − βN−1)] − V y
}
. (66)
In equation (66) it was found that
φ =
u − w − V
u − w
(1 − ξ)
[
N − 1 − 1 − βN−1
1 − β
]
+
u − w + V
u − w
ξ
[
(N − 1)βN−1 − β
1 − βN−1
1 − β
]
(67)
and F is given by (64).
Comparison between (65) and (14) shows that they are identical (with replacement R → T ),
with the exception of χyk term in (65). This implies that
T
(1)
kN = B̃yk (68)
with arbitrary constant B̃ solves homogeneous equation [equation (65) without χyk term], since y
given by (23) is constructed for RkN = R0y
k to solve specifically equation (14).
To account for χyk term, we look for solution of (65) of the form
T
(2)
kN = C̃zk, (69)
where
z =
−b−
√
b2 − 4ac
2c
(70)
with a, b, c given by (15)–(17); parameter C̃ is to be determined. Parameter z is the solution of
equation cz2 + bz + a = 0 such that 0 6 z < 1. Plugging (69) into (65) gives
C̃zk(a + bz + cz2) + (d + ey + fy2)C̃yk + χyk = 0. (71)
The first term in (71) vanishes since z is given by (70), thus (71) yields
C̃ =
−χ
d + ey + fy2
. (72)
The T
(2)
kN given by (69) with C̃ specified by (72) therefore solves equation (65). More generally, (65)
is solved by the sum of contributions (68) and (69), i. e.
TkN = B̃yk + C̃zk, (73)
23801-9
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
where B̃ remains undetermined. Plugging (73) in equation (63) for θk gives
θk =
1 − β
1 − βN
{
B̃yk(y − 1) + C̃zk(z − 1) − FR0y
k + p2y
k(1 − y)[B̃ + C̃ ]
}
. (74)
Substituting (73) and (74) into (59) results in the final expression for TkN+i,
TkN+i = B̃yk + C̃zk +
1 − βi
1 − βN
{
B̃yk(y − 1) + C̃zk(z − 1) − FR0y
k + p2y
k(1 − y)[B̃ + C̃]
}
+ R0y
k
{
u − w − V
u − w
(1 − ξ)
[
i − 1 − βi
1 − β
]
+
u − w + V
u − w
ξ
[
iβi − β
1 − βi
1 − β
]}
. (75)
Although equation (45) was not utilized to obtain (75), it was numerically checked that every
equation in the system (45)–(48) holds for TkN+i, RkN+i given by (75) and (25).
To compute the diffusion coefficient D, we need to consider additional auxiliary functions [7, 10],
UkN+i(t) =
+∞
∑
m=−∞
(mN + kN + i)2PmN+kN+i,m(t), k > 0, i = 0, 1, · · · , N − 1. (76)
A system of equations which determine the time evolution of Uj(t) is derived with the use of
equations (76), (34) and (33) for UkN+i(t), SkN+i(t) and RkN+i(t), as well as equations (29)–(32).
It follows that
dU0(t)
dt
= wU1(t) − (u + p2w)U0(t) − 2wS1(t) + wR1(t) + p1u
∞
∑
α=1
UαN−1(t)
+ 2p1u
∞
∑
α=1
SαN−1(t) + p1u
∞
∑
α=1
RαN−1(t), (77)
dUkN+(N−1)(t)
dt
= wU(k+1)N (t) + uUkN+(N−2)(t) − (u + w)UkN+(N−1)(t)
+ p2wf(k + 1)U0(t) − 2wS(k+1)N (t) + 2uSkN+(N−2)(t)
− 2p2wf(k + 1)S0(t) + wR(k+1)N (t) + uRkN+(N−2)(t)
+ p2wf(k + 1)R0(t) (78)
for k > 0,
dUkN (t)
dt
= wUkN+1(t) + (1 − p1)uUkN−1(t) − (u + w)UkN (t) − 2wSkN+1(t)
+ 2(1− p1)uSkN−1(t) + wRkN+1(t) + (1 − p1)uRkN−1(t) (79)
for k > 1, and
dUkN+i(t)
dt
= wUkN+i+1(t) + uUkN+i−1(t) − (u + w)UkN+i(t) − 2wSkN+i+1(t)
+ 2uSkN+i−1(t) + wRkN+i+1(t) + uRkN+i−1(t) (80)
for k > 0 and i = 1, · · · , N − 2.
The diffusion constant is to be found from
D =
1
2
lim
t→∞
d
dt
[
〈x(t)2〉 − 〈x(t)〉2
]
. (81)
23801-10
Reversible burnt-bridge models
Using (76) and summing up equations (77)–(80) results in
d
dt
〈x(t)2〉 =
d
dt
+∞
∑
m=−∞
∞
∑
k=0
N−1
∑
i=0
(mN + kN + i)2PmN+kN+i,m(t)
=
d
dt
∞
∑
k=0
N−1
∑
i=0
UkN+i(t) =
∞
∑
k=0
N−1
∑
i=0
dUkN+i(t)
dt
= (u + w) − w(1 − p2)R0(t) + 2w(1 − p2)S0(t) + 2(u − w)
∞
∑
k=0
N−1
∑
i=0
SkN+i(t), (82)
thus we do not need to find UkN+i(t) from the system (77)–(80) to obtain the diffusion coefficient.
To derive (82) we used the normalization conditions,
∞
∑
k=0
N−1
∑
i=0
RkN+i(t) = 1 and
∞
∑
k=0
f(k + 1) = 1.
In the t → ∞ limit, SkN+i(t) → akN+it + TkN+i, R0(t) → R0, thus (82) becomes
d
dt
〈x(t)2〉 = (u+w)−w(1−p2)R0 +2w(1−p2)[a0t+T0]+2(u−w)
∞
∑
k=0
N−1
∑
i=0
[akN+it+TkN+i]. (83)
Given that in the stationary-state limit d
dt
〈x(t)〉 =
∞
∑
k=0
N−1
∑
i=0
akN+i = w(1 − p2)R0 + (u − w) = V
[equation (53)] and utilizing 〈x(t)〉 given by (51), it follows that
d
dt
〈x(t)〉2 = 2〈x(t)〉 d
dt
〈x(t)〉 = 2V 〈x(t)〉 = 2V
∞
∑
k=0
N−1
∑
i=0
SkN+i(t → ∞)
= 2[w(1 − p2)R0 + (u − w)]
∞
∑
k=0
N−1
∑
i=0
{akN+it + TkN+i} . (84)
Plugging (83) and (84) into (81) gives the diffusion constant,
D=
1
2
[
(u + w) − w(1 − p2)R0 + 2w(1 − p2)[a0t + T0] − 2w(1 − p2)R0
∞
∑
k=0
N−1
∑
i=0
{akN+it + TkN+i}
]
.
(85)
The time-dependent part of (85) is
D̃(t) =
1
2
[
2w(1 − p2)a0t − 2w(1 − p2)R0
∞
∑
k=0
N−1
∑
i=0
akN+it
]
. (86)
Using
∞
∑
k=0
N−1
∑
i=0
akN+i = V [equation (53)] and a0 = V R0 [equation (50)], it follows that according
to (86) D̃(t) = 0. Therefore, as anticipated, the diffusion constant does not contain time-dependent
terms, and it is given by
D =
1
2
[
(u + w) − w(1 − p2)R0 + 2w(1 − p2)T0 − 2w(1 − p2)R0
∞
∑
k=0
N−1
∑
i=0
TkN+i
]
. (87)
In order to get the final expression for diffusion coefficient, it is necessary to calculate
∞
∑
k=0
N−1
∑
i=0
TkN+i
in (87). Using equation (75) for TkN+i yields,
∞
∑
k=0
N−1
∑
i=0
TkN+i =
B̃N
1 − y
+
C̃N
1 − z
+
[
N
1 − βN
− 1
1 − β
]{
−B̃ − C̃ − R0F
1 − y
+ p2(B̃ + C̃)
}
+ λ, (88)
23801-11
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
where
λ =
R0
1 − y
{
u − w − V
u − w
(1 − ξ)
[
N(N − 1)
2
− 1
1 − β
(
N − 1 − βN
1 − β
)]
+
u − w + V
u − w
ξ
[−NβN + (N − 1)β(N+1) + β
(1 − β)2
− β
1 − β
(
N − 1 − βN
1 − β
)]}
. (89)
In deriving (88), we used the fact that 0 6 y < 1, 0 6 z < 1. Next, we consider the last two terms
in (87),
2w(1 − p2)T0 − 2w(1 − p2)R0
∞
∑
k=0
N−1
∑
i=0
TkN+i = 2w(1 − p2)
{
B̃ + C̃ − R0
(
B̃N
1 − y
+
C̃N
1 − z
+
[
N
1 − βN
− 1
1 − β
] [
−B̃ − C̃ − R0F
1 − y
+ p2(B̃ + C̃)
]
+ λ
)}
, (90)
where we used (88) and T0 = B̃ + C̃ [equation (73)]. The contribution from terms ∝ B̃ in (90) can
be shown to be
2w(1 − p2)B̃
{
1 − R0
1− y
(
N − (1 − p2)(1 − y)
[
N
1 − βN
− 1
1 − β
])}
. (91)
Utilizing equation (26) for R0, it follows from (91) that B̃-contribution in (90) equals 0, thus
undetermined constant B̃ cancels out in (87) and it has no effect on the diffusion coefficient.
Without B̃-terms equation (90) becomes
2w(1 − p2)
[
T0 − R0
∞
∑
k=0
N−1
∑
i=0
TkN+i
]
= 2w(1 − p2)
{
C̃ − R0
(
C̃N
1 − z
−
[
N
1 − βN
− 1
1 − β
] [
R0F
1 − y
+ (1 − p2)C̃
]
+ λ
)}
. (92)
Plugging (92) into (87) gives
D =
1
2
{
(u + w) − w(1 − p2)R0 + 2w(1 − p2)C̃ − 2w(1 − p2)R0
(
C̃N
1 − z
−
[
N
1 − βN
− 1
1 − β
] [
R0F
1 − y
+ (1 − p2)C̃
]
+ λ
)}
. (93)
In equation (93) we have β = u/w, and parameters R0, C̃, y, z, F , λ are given by equations (26),
(72), (23), (70), (64), (89), correspondingly. Other useful parameters [on which D in (93) implicitly
depends] are a − f , χ, ξ, V given by equations (15)–(20), (66), (54), and (28), correspondingly.
With the help of these parameters equation (93) gives the exact expression for D as a function of
transition rates u, w, probabilities p1, p2, and concentration of bridges c = 1/N .
It was verified numerically for various p1 and c values that in the limit of u → 1, w → 1,
p2 → 0 equation (93) reproduces the diffusion constant obtained in [7] for BBM with u = w = 1
and p2 = 0. In the limiting case of p1 = 1 we obtained D(u, w, p2, N) in [9] using Derrida method
[10] and it also agrees with our general result (93) in the p1 → 1 limit, as was numerically checked.
4. Discussions
To illustrate our findings, we plot the dynamic properties of the molecular motor using equati-
ons (28) and (93). We first consider the case of the unbiased molecular motor with transition rates
23801-12
Reversible burnt-bridge models
(a) (b)
Figure 3. Dynamic properties of the unbiased molecular motor with u = w = 1 and with the
probability of burning p1 = 0.1: (a) The mean velocity as a function of the concentration of
bridges for different recovery probabilities; (b) the dispersion as a function of the concentration
of bridges for different recovery probabilities.
(a) (b)
Figure 4. Dynamic properties of the unbiased molecular motor with u = w = 1 and with the
recovery probability p2 = 0.4: (a) The mean velocity as a function of the concentration of bridges
for different burning probabilities; (b) the dispersion as a function of the concentration of bridges
for different burning probabilities.
(a) (b)
Figure 5. Dynamic properties of the unbiased molecular motor with u = w = 1 and with the
concentration of bridges c = 0.2: (a) The mean velocity as a function of the burning probability
for different recovery probabilities; (b) dispersion as a function of the burning probability for
different recovery probabilities.
23801-13
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
u = w = 1 (figures 3–5). Equations (28) and (93) cannot be used directly when β = u/w = 1.
Whereas it is possible to find the u → 1, w → 1 limit of equation (28) (although it leads to a cum-
bersome expression) and thus to plot the velocity, it is problematic in the case of equation (93).
Thus we plotted the diffusion constant for u = 0.999 and w = 1 (see figures 3 (b), 4 (b), 5 (b)).
Using u values closer to 1 generates numerical instability. This is a good approximation of the
u = w = 1 case, as was judged from comparison with known limiting cases. Namely, comparing
p2 = 0 case [figures 3 (b), 5 (b)] with the result from [7] for u = w = 1 showed the discrepancy in
D values of the order of ' 0.001 for almost entire range of parameters c and p1, with the exception
of small c . 0.02, and p1 . 0.01, where discrepancy exceeded 0.008 and 0.006 correspondingly. For
figure 4 (b), we compared p1 = 1 case (not shown) with the corresponding case for u = w = 1
obtained in [9]: typical discrepancy in D values between u = 0.999 and u = 1 cases was ∼ 0.0005
for all c values except c . 0.001, where discrepancy exceeded 0.007.
As anticipated, when the recovery probability p2 → 1 and the presence of bridges has no effect,
V → u−w = 0 and D → 1
2 (u+w) = 1 for all c (and p1) values (figures 3 and 5); the same happens
in the limit of the burning probability p1 → 0 (figure 4). Increasing p1 and the concentration of
weak links c leads to increasing velocity [as shown in figures 3 (a), 4 (a), 5 (a)], whereas increasing
p2 reduces the velocity [figures 3 (a), 5 (a)].
(a) (b)
Figure 6. Dynamic properties of the backward biased molecular motor with u = 0.3 and w =
0.7, and with the burning probability p1 = 0.3: (a) The mean velocity as a function of the
concentration of bridges for different recovery probabilities; (b) the dispersion as a function of
the concentration of bridges for different recovery probabilities.
The diffusion constant plotted in figures 3 (b) and 4 (b) is a decreasing function of the bridge
concentration c because the presence of bridges lowers the fluctuations of the motor protein (the
motor protein cannot cross back the already burned bond). We observed that in the limit of low
c there is a gap in the dispersion [figures 3 (b) and 4 (b)]: D(c → 0) differs from the expected
D(c = 0) value of 1
2 (u + w) = 1. This phenomenon corresponds to a dynamic transition between
unbiased and biased diffusion regimes as was argued earlier in [9]. Figure 3(b) is similar to the
corresponding plot in [9] for p1 = 1 case, but for p1 = 0.1 the gap is prominent only for small p2
values, and for p2 > 0.1 it practically disappears. We note that for p2 = 0 case in figure 3 (b) the
correct c → 0 limit must be D(c → 0) = 2/3 [7]. However, it did not reach this value because of
the emerging numerical instability for very low c . 0.001 (see also our discussion above).
Analysis of figure 5 (b) shows that as p2 increases, the behavior of diffusion constant changes
from increasing to decreasing function of p1, with D(p1) developing a minimum for p2 . 0.1. It
should be noted that in p2 = 0 case, D should approach the value of 1/2 for p1 → 0 according
to [7]. We see some discrepancy there which is also due to the numerical instability for small p1
values. In addition, we observed a gap between D(p1 → 0) = 1/2 for p2 = 0 and D(p1 → 0) = 1
for all nonzero p2 values.
As another example, we investigated the case of the backward biased motor protein (with
specific transition rates u = 0.3, w = 0.7), where bridges are inducing the molecular motor to
23801-14
Reversible burnt-bridge models
(a) (b)
Figure 7. Dynamic properties of the backward biased molecular motor with u = 0.3 and w =
0.7, and with the recovery probability p2 = 0.1: (a) The mean velocity as a function of the
concentration of bridges for different burning probabilities; (b) the dispersion as a function of
the concentration of bridges for different burning probabilities.
(a) (b)
Figure 8. Dynamic properties of the backward biased molecular motor with u = 0.3 and w =
0.7, and with the recovery probability p2 = 0.6: (a) The mean velocity as a function of the
concentration of bridges for different burning probabilities; (b) the dispersion as a function of
the concentration of bridges for different burning probabilities.
(a) (b)
Figure 9. Dynamic properties of the backward biased molecular motor with u = 0.3 and w = 0.7,
and with the concentration of bridges c = 0.2: (a) The mean velocity as a function of the burning
probability for different recovery probabilities; (b) the dispersion as a function of the burning
probability for different recovery probabilities.
23801-15
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
move in the opposite direction (figures 6–9). The analysis of dynamic properties shows that in the
p2 → 1 limit (the deterministic bridge recovery) V → u − w = −0.4 and D → 1
2 (u + w) = 1/2
for all c and p1 values (see figures 6 and 9), as it should be. The p1 → 0 limit in the u < w case
is more complex than in the u > w case (when V → u − w, D → 1
2 (u + w) for all p2 6= 0 and
c values as p1 → 0). Namely, for u < w there exists p̃2(u, w) such that for non-zero p2 < p̃2 the
dynamic properties V (c) and D(c) exhibit strong c-dependence in the p1 → 0 limit and they are
different from the u − w and 1
2 (u + w) values for non-zero c [figure 7 (a), (b)]. Exactly at p1 = 0
(with p2 < p̃2), V (p1 = 0, c) = u − w and D(p1 = 0, c) = 1
2 (u + w) as obtained using the y = 1
root of equation (22), given by equation (23) with “+” rather than “−” sign before the square
root [the second root of equation (22)]. Thus for p2 < p̃2 there is a dynamic transition separating
p1 = 0 and p1 → 0 regimes. For u = 0.3, w = 0.7, it was found that p̃2 ≈ 0.57. For p2 > p̃2,
V (p1 → 0, c) = V (p1 = 0, c) = u − w = −0.4 and D(p1 → 0, c) = D(p1 = 0, c) = 1
2 (u + w) = 0.5
for all c values [figure 8 (a), (b)]. Physically this implies that bridges with arbitrarily low burning
probability strongly affect the dynamics of the particle which tends to move in the backward
direction provided that the recovery probability p2 is less than some critical value; otherwise (in
p2 > p̃2 case) weak links have no effect on the particle dynamics as p1 → 0. Thus in the p1 → 0
limit there is a dynamic transition at p2 = p̃2 separating the regime with p2 < p̃2 when weak links
play a role in the motor protein dynamics and the regime where they are irrelevant (p2 > p̃2). It
should be noted that the p1 → 0 limit in figure 9 is in agreement with that in figures 7 and 8:
p̃2 ≈ 0.57 plays the same role. The dynamic transition in the p1 → 0 limit also takes place for
p2 = 0 (irreversible burning of weak connections). It separates backward biased p1 = 0 regime from
forward biased regime with small finite p1, when velocity V (c) is positive for all c > 0, although
V (c) → 0, D(c) → 0 for all c values as p1 → 0 [see figure 9 (a), (b) with the specific c value]. There
are therefore jumps in V (c) and D(c) at p1 = 0 for all p2 < p̃2 and all non-zero c.
Increasing the burning probability p1 and concentration of weak bonds c reduces the magnitude
of particle’s velocity in the backward direction; the same effect is observed when the recovery
probability p2 is reduced, as expected [see figures 6 (a), 7 (a), 8 (a), 9 (a)]. For sufficiently large c,
p1 (and small p2) the velocity V even becomes positive [figures 6 (a), 7 (a)]. We observed that for
p2 = 0, V is always positive as the burning of weak bonds is irreversible in this case [figures 6 (a),
9 (a)]. In that case, V (c = 0) = u−w is different from V (c → 0) = 0 [figure 6 (a)]. This effect was
also observed in [9] in the p1 = 1 case.
Diffusion constant demonstrates a more complex behavior, with large fluctuations at small c
and small p2 which increase with increasing p1 [figures 6 (b) and 7 (b)]. Figure 6(b) with p1 = 0.3
is qualitatively similar to the corresponding figure in [9] with p1 = 1, although the maxima in
D curves are less pronounced than in [9]. The shape of the D curve differs significantly between
figures 7 (b) and 8 (b) when the threshold p̃2 is crossed. In case of p2 = 0 (irreversible bridge
burning), fluctuations are reduced (especially at low c) and D curve differs substantially from non-
zero p2 case [figures 6 (b) and 9(b)]. As was the case with the velocity, for p2 = 0 there is a gap in
D separating D(c = 0) = 1
2 (u + w) from D(c → 0) = 0 [figure 6 (b)], again illustrating a dynamic
transition. For non-zero p2 the diffusion constant is D(c → 0) = 1
2 (u + w), and there is no gap [see
figures 6 (b), 7 (b), 8 (b)].
5. Conclusions
We have presented a comprehensive theoretical method of calculating dynamic properties of
molecular motors in reversible burnt-bridge models for periodic bridge distribution. It is a generali-
zation of the approach developed by us in [6, 7] for the unbiased molecular motors and irreversible
burning of bridges. Exact and explicit expressions for mean velocity and dispersion have been de-
rived for arbitrary values of parameters u, w, p1, p2 and c. In the known limiting cases of u = w = 1,
p2 = 0 and of p1 = 1, we have reproduced our earlier findings [6, 7, 9], thereby confirming the
validity of our theoretical analysis. Some interesting phenomena have been observed as a result of
the investigation of dynamic properties of the molecular motor in BBM with bridge recovery. It in-
cludes dynamic phase transitions and reversal of the direction of the motion. In case of the unbiased
23801-16
Reversible burnt-bridge models
molecular motor, increasing the concentration of bridges c (or lowering the recovery probability p2)
with other parameters kept fixed results in increasing velocity and decreasing dispersion. However,
dependence of the dispersion on burning probability p1 is more complex; it is determined by the
p2 value. In the limit of low c, gaps in dispersion plots have been observed for various p1 and p2
values, indicating the dynamic transition between biased and unbiased regimes. Also, a gap was
found in the limit of small p1 between p2 = 0 and non-zero p2 regimes. Thus our results obtained
in [9] for p1 = 1 with u = w were generalized to cover the full range of p1 values.
For the backward biased molecular motor, increasing c has resulted in slowing down the back-
ward movement of the particle, and for sufficiently small p2 (large p1) the direction of motion has
even been reversed and the velocity became positive. In the limit of small p1, a dynamic phase
transition separating p1 = 0 and p1 → 0 regimes has been found provided that p2 is less than
some critical value. For sufficiently small p2, broken bridges affect the particle’s dynamics even
if the burning probability p1 is infinitesimal. The behavior of dispersion as a function of c was
non-monotonic for some range of parameters p1 and p2, with large fluctuations at small c and
small p2. In the case of irreversible bridge burning (p2 = 0), we have observed gaps in velocity and
dispersion in c → 0 limit (for p1 = 0.3), with the velocity being positive for all non-zero c values. It
suggests that there is a dynamic transition at c = 0 separating backward biased and forward biased
diffusion. The velocity and fluctuations are suppressed for sufficiently small c. Hence our findings
in [9] for u < w case with p1 = 1 have been extended to describe the general case of 0 < p1 6 1.
The method presented above applies to the case of periodic distribution of weak bonds, which
is probably realistic for collagenases [2]. As a problem to be addressed in the future studies, one
can consider BBM with random distribution of bridges [4, 5] where a different theoretical approach
must be applied.
References
1. Kolomeisky A.B., Fisher M.E., Ann. Rev. Phys. Chem., 2007, 58, 675.
2. Saffarian S., Collier I.E., Marmer B.L., Elson E.L., Goldberg G., Science, 2004, 306, 108.
3. Saffarian S., Qian H., Collier I.E., Elson E.L., Goldberg G., Phys. Rev. E, 2006, 73, 041909.
4. Mai J., Sokolov I.M., Blumen A., Phys. Rev. E, 2001, 64, 011102.
5. Antal T., Krapivsky P.L., Phys. Rev. E, 2005, 72, 046104.
6. Morozov A.Y., Pronina E., Kolomeisky A.B., Artyomov M.N., Phys. Rev. E, 2007, 75, 031910.
7. Artyomov M.N., Morozov A.Y., Pronina E., Kolomeisky A.B., J. Stat. Mech., 2007, P08002.
8. Morozov A.Y., Kolomeisky A.B., J. Stat. Mech., 2007, P12008.
9. Artyomov M.N., Morozov A.Y., Kolomeisky A.B., Phys. Rev. E, 2008, 77, 040901(R).
10. Derrida B., J. Stat. Phys., 1983, 31, 433.
23801-17
M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky
Динамiка молекулярних двигунiв у оборотнiй моделi
спалених мостiв
М.Н. Артьомов1, А.Ю. Морозов2, А.Б. Коломейскi3
1 Хiмiчний факультет, Массачусеттський технологiчний iнститут, Кембрiдж, США
2 Факультет фiзики i астрономiї, Унiверситет Калiфорнiї, Лос Анжелес, США
3 Хiмiчний факультет, Унiверситет Райса, Г’юстон, США
Динамiчнi властивостi молекулярних двигунiв, чий рух здiйснюється завдяки взаємодiям з особли-
вими ґратковими зв’язками, дослiджуються теоретично на основi стохастичних моделей “спалених
мостiв” з дискретними станами. Молекулярнi двигуни описуються як випадковi блукання, що можуть
знищувати чи вiдновлювати перiодично розподiленi зв’язки (“мости” при їх проходженнi з ймовiрно-
стями, вiдповiдно, p1 та p2. Динамiчнi властивостi, такi як швидкостi та дисперсiї, отримуються в
точнiй та явнiй формi для довiльних значень параметрiв p1 i p2. Для неупередженого випадкового
блукання оборотне спалення мостiв приводить до упередженого спрямованого руху з динамiчним
переходом, що спостерiгається при дуже малих концентрацiях мостiв. У випадку упередженого мо-
лекулярного двигуна, що може рухатись назад, його швидкiсть зменшується та змiна напрямку руху
спостерiгається для деякої областi параметрiв. Отримано також, що дисперсiя демонструє складну
немонотонну поведiнку з великими флуктуацiями для деякого набору параметрiв. Складна динамi-
ка системи обговорюється за допомогою аналiзу поведiнки молекулярних двигунiв бiля спалених
мостiв.
Ключовi слова: молекулярнi двигуни, стохастичнi моделi, рухливi протеїни
23801-18
Introduction
Model
Dynamic properties for BBM with bridge recovery
Velocity
Diffusion coefficient
Discussions
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-32101 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:51:12Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Artyomov, M.N. Morozov, A.Yu. Kolomeisky, A.B. 2012-04-08T16:32:59Z 2012-04-08T16:32:59Z 2010 Dynamics of molecular motors in reversible burnt-bridge models / M.N. Artyomov, A.Yu. Morozov, A.B. Kolomeisky // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23801: 1-18. — Бібліогр.: 10 назв. — англ. 1607-324X PACS: 87.16.Ac https://nasplib.isofts.kiev.ua/handle/123456789/32101 Dynamic properties of molecular motors whose motion is powered by interactions with specific lattice bonds are studied theoretically with the help of discrete-state stochastic "burnt-bridge" models. Molecular motors are depicted as random walkers that can destroy or rebuild periodically distributed weak connections ("bridges") when crossing them, with probabilities p1 and p2 correspondingly. Dynamic properties, such as velocities and dispersions, are obtained in exact and explicit form for arbitrary values of parameters p1 and p2. For the unbiased random walker, reversible burning of the bridges results in a biased directed motion with a dynamic transition observed at very small concentrations of bridges. In the case of backward biased molecular motor its backward velocity is reduced and a reversal of the direction of motion is observed for some range of parameters. It is also found that the dispersion demonstrates a complex, non-monotonic behavior with large fluctuations for some set of parameters. Complex dynamics of the system is discussed by analyzing the behavior of the molecular motors near burned bridges. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Dynamics of molecular motors in reversible burnt-bridge models Динаміка молекулярних двигунів у оборотній моделі спалених мостів Article published earlier |
| spellingShingle | Dynamics of molecular motors in reversible burnt-bridge models Artyomov, M.N. Morozov, A.Yu. Kolomeisky, A.B. |
| title | Dynamics of molecular motors in reversible burnt-bridge models |
| title_alt | Динаміка молекулярних двигунів у оборотній моделі спалених мостів |
| title_full | Dynamics of molecular motors in reversible burnt-bridge models |
| title_fullStr | Dynamics of molecular motors in reversible burnt-bridge models |
| title_full_unstemmed | Dynamics of molecular motors in reversible burnt-bridge models |
| title_short | Dynamics of molecular motors in reversible burnt-bridge models |
| title_sort | dynamics of molecular motors in reversible burnt-bridge models |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32101 |
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