Lamplighter model of a random copolymer adsorption on a line
We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of t...
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2014
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/153509 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Lamplighter model of a random copolymer adsorption on a line / L.I. Nazarov, S.K. Nechaev, M.V. Tamm // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33002:1-8. — Бібліогр.: 12 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-153509 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1535092019-06-15T01:27:26Z Lamplighter model of a random copolymer adsorption on a line Nazarov, L.I. Nechaev, S.K. Tamm, M.V. We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of the local contact energy minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total energy, E (the number of unfavorable contacts), on the number of chain monomers, N: E ~ N³/⁴ for quenched random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping it onto the "lamplighter" random walk and the diffusion-controlled chemical reaction of X+X → 0 type with the subdiffusive motion of reagents. Ми представляємо модель послiдовного самопакування AB-диблокового випадкового кополiмера з локальними замороженими взаємодiями на одновимiрнiй нескiнченiй липкiй основi. Припускається, що контакти A-A i B-B є сприятливi, тодi як A-B є несприятливими. Положення нового мономера, що додається, виберається з точки зору мiнiмiзацiї енергiї локального контакту. Модель демонструє саморганiзовану поведiнку з нетривiальною залежнiстю загальної енергiї, E (числа несприятливих контактiв), вiд числа мономерiв ланцюга, N: E ∼ N ³/⁴ для замороженого хаотичного рiвноймовiрного розподiлу Ai B-мономерiв вздовж ланцюга. Модель розглядається шляхом зiставлення її з випадковим блуканням лiхтарника i дифузiйно-контрольованою хiмiчною реакцiєю типу X + X → 0 з субдифузiйним рухом реагентiв. 2014 Article Lamplighter model of a random copolymer adsorption on a line / L.I. Nazarov, S.K. Nechaev, M.V. Tamm // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33002:1-8. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 05.40.-a, 02.50.Ga, 87.15.Cc DOI:10.5488/CMP.17.33002 arXiv:1405.0380 http://dspace.nbuv.gov.ua/handle/123456789/153509 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of the local contact energy minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total energy, E (the number of unfavorable contacts), on the number of chain monomers, N: E ~ N³/⁴ for quenched random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping it onto the "lamplighter" random walk and the diffusion-controlled chemical reaction of X+X → 0 type with the subdiffusive motion of reagents. |
| format |
Article |
| author |
Nazarov, L.I. Nechaev, S.K. Tamm, M.V. |
| spellingShingle |
Nazarov, L.I. Nechaev, S.K. Tamm, M.V. Lamplighter model of a random copolymer adsorption on a line Condensed Matter Physics |
| author_facet |
Nazarov, L.I. Nechaev, S.K. Tamm, M.V. |
| author_sort |
Nazarov, L.I. |
| title |
Lamplighter model of a random copolymer adsorption on a line |
| title_short |
Lamplighter model of a random copolymer adsorption on a line |
| title_full |
Lamplighter model of a random copolymer adsorption on a line |
| title_fullStr |
Lamplighter model of a random copolymer adsorption on a line |
| title_full_unstemmed |
Lamplighter model of a random copolymer adsorption on a line |
| title_sort |
lamplighter model of a random copolymer adsorption on a line |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153509 |
| citation_txt |
Lamplighter model of a random copolymer
adsorption on a line / L.I. Nazarov, S.K. Nechaev, M.V. Tamm // Condensed Matter Physics. — 2014. — Т. 17, № 3. — С. 33002:1-8. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT nazarovli lamplightermodelofarandomcopolymeradsorptiononaline AT nechaevsk lamplightermodelofarandomcopolymeradsorptiononaline AT tammmv lamplightermodelofarandomcopolymeradsorptiononaline |
| first_indexed |
2025-07-14T04:38:36Z |
| last_indexed |
2025-07-14T04:38:36Z |
| _version_ |
1837595798727032832 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 3, 33002: 1–8
DOI: 10.5488/CMP.17.33002
http://www.icmp.lviv.ua/journal
Lamplighter model of a random copolymer
adsorption on a line
L.I. Nazarov1, S.K. Nechaev2,3,4, M.V. Tamm1,4
1 Physics Department, M.V. Lomonosov Moscow State University, 119991 Moscow, Russia
2 Université Paris-Sud/CNRS, LPTMS, UMR8626, Bât. 100, 91405 Orsay, France
3 P.N. Lebedev Physical Institute, RAS, 119991 Moscow, Russia
4 Department of Applied Mathematics, National Research University “Higher School of Economics”,
101000 Moscow, Russia
Received April 30, 2014
We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interac-
tions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable,
while A-B are not. The position of a newly added monomer is selected in view of the local contact energy
minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total
energy, E (the number of unfavorable contacts), on the number of chain monomers, N : E ∼ N 3/4 for quenched
random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping
it onto the “lamplighter” random walk and the diffusion-controlled chemical reaction of X + X → 0 type with
the subdiffusive motion of reagents.
Key words: local optimization, heteropolymer folding, lamplighter random walk, subdiffusive chemical
reaction
PACS: 05.40.-a, 02.50.Ga, 87.15.Cc
1. Introduction
In this letter we propose a simple one-dimensional model of stochastic dynamic system possessing a
local optimization in quenched environment. The question of choosing an optimal strategy for a dynamic
system if only local optimization is accessible, is a generic problem far beyond the scope of natural sci-
ence. Let us imagine that someone knows nothing about the future and adapts his or her own behavior in
each current time moment only on the basis of the knowledge about the best solution at a given narrow
time slice. Such a behavior leads to an optimal local strategy when only a partial (current) knowledge is
accessible, though it might be far from generic optimal one if the knowledge about the future is available.
As an example of physical problems sharing the properties of local and global optimization, we can
mention the problems of protein secondary structure formation and DNA packaging in a viral capsid. If in
the course of protein folding, the chain forms a “frozen” network of contacts, then the optimal conforma-
tion is determined only by a sequential step-by-step optimization of the added heteropolymer fragments.
However, if the whole protein is capable of “adjusting” the structure of a contact network by breaking
some bonds and creating other bonds, then the global optimization might essentially change the folding
picture.
The system considered below should not be regarded as a model of any specific physical system
(though it has some features of a protein folding), but it rather highlights the principles of local opti-
mization of a specific one-dimensional stochastic system in quenched random environment, which could
lead to a nontrivial self-organization.
Our model resembles the “lamplighter random walk” proposed in 1973 by A.M. Vershik in the after-
word to the Russian edition [1] of the book “Invariantmeans on topological groups and their applications”
© L.I. Nazarov, S.K. Nechaev, M.V. Tamm, 2014 33002-1
http://dx.doi.org/10.5488/CMP.17.33002
http://www.icmp.lviv.ua/journal
L.I. Nazarov, S.K. Nechaev, M.V. Tamm
by F.P. Greenleaf as an example of nontrivial estimate for the growth of various numerical characteris-
tics of groups. The object known as a “lamplighter group” (the name was also given by A.Vershik) became
very popular among probabilists after the pioneering work by A. Vershik and V. Kaimanovich [2].
2. The model
The toymodel presented below imitates the packaging of a random diblock copolymer with quenched
sequence of monomers on a one-dimensional substrate. Schematically it can be viewed as a tissue folding
getting out of the roll, as depicted in figure 1 (a), where the size of folds depends on local heterogenous
interactions between different parts of the tissue. Such a schematic view might be compared with the
specific secondary structure formation of a linear polypeptide chain in figure 1 (b), where two main
types of secondary structure, α-helices and β-sheets, are shown and the secondary structure is stabilized
by the hydrogen bonds.
Figure 1. (Color online) (a) Folds of tissue getting out of the roll; (b) Secondary structure formation in a
polypeptide heteropolymer chain.
To pass from these intuitive pictures to amore specific description, consider aN -monomer chain with
quenched random primary sequence of two types of links, A and B, placed on an infinite line, which is
sticky for links A and repulsive for links B. We suppose that the energy of the A-A or B-B contacts is 0,
while the energy of the A-B contact is +1 (in dimensionless units), i.e., the monomers of the same types
are indifferent to each other, while the monomers of different types repulse.
The heteropolymer folding (packaging) on an infinite line is an irreversible sequential process of
adding monomers to the already existing frozen environment. Let us recursively describe the folding
process. For definiteness, suppose that the line is initially uniformly covered with monomers of type A
and let the first chain monomer be always of type A. Put the 1st monomer at a position x0 = 0 on the line.
The 2nd monomer could be placed either next to the 1st monomer on a line in the trans state, or it can be
put on top of the 1st monomer in the gauche state, making a “hook” as it is shown in figure 2. The newly
added monomer interacts with the one located below it in the projection to the line.
Figure 2. (Color online) Local rules of the monomers freezing to the structure. (a) The monomer 3 has two
possible locations I and II. The selection of the position depends on the types of all three monomers. (b)
Monomers 2 and 3 are identical, while the monomer 1 differs from 2 and 3. (c) The inverse situation with
respect to (b).
33002-2
Lamplighter model of a copolymer adsorption
Figure 3. (Color online) The 20-step of block copolymer package on the plate with the uniform monomer
type distribution [(a) The side view; (b) The top view].
The selection between trans- and gauche- states is made according to the following rules: i) if one
of two possible new (trans- or gauche-) states is favorable and the other is not, then the favorable state
is always selected; ii) if new trans- and gauche- states are both favorable or both unfavorable, then a
new conformation (trans- or gauche-) is chosen randomly with the probability 1/2. For any “defect” (i.e.,
unfavorable pairing) the energy penalty+1 is added. The sample structure of the first 20 steps is depicted
in figure 3.
The position of a newly added monomer is selected in view of the irreversible local contact energy
minimization, while for equal energies, the choice of trans- or gauche- states is randomwith equal proba-
bilities. Let us emphasize that only the uppermost monomers or the “roof” of the structure [i.e., the ones
visible from the top of the picture in figure 3 (b)] take part in the play: the monomers that are underneath
do not participate in a further optimization process. Therefore, the folding is an effective Markovian pro-
cess, whose states are specified by i) the set of monomers visible currently from the top of the structure
(i.e., by the “roof”), and ii) the position and direction of the last step, which defines the positions where
the next monomer can be added.
3. Results
In this section we consider the properties of the described locally optimal folding averaged over a set
of completely random initial primary sequences of a polymer, i.e., we assume A and B monomers to ap-
pear in the chain with equal probabilities without any correlations between sequential letters. The most
natural question to ask about this folding is how the average total energy penalty, 〈E〉, depends on the
chain length, N , i.e., how many unfavorable A-B contacts one has on average in the N -monomer chain.
Unexpectedly, it turns out that 〈E〉 grows sub-linearly withN , namely as 〈E〉 ∼ N 3/4
[see figure 5 (center)].
To understand this result, one should notice that in terms of the evolution of the roof of the structure
defined above, the system has an adsorbing state. Indeed, if the roof is represented by an alternating
sequence, i.e., . . . -A-B-A-B-. . . , every newly added monomer can be positioned on the top of this periodic
sequencewithout any energetic penalty, and the resulting structure of the roof will stay unchanged. More-
over, if some part of the roof is alternating [as in figure 3 (b)], this part will remain alternating and any
sequence can be adsorbed on it without penalty. Thus, the heteropolymer folding in our model can be
considered as a dynamic process of approaching the adsorbing alternating state. This dynamics consists
in sequential cancellation of “defects”, represented by non-alternating pairs of A-A or B-B of consecutive
monomers on the substrate as it is shown in figure 4 (a).
To formalize this idea, we introduce an Ising variable, s(x), which labels the state of the roof in posi-
tion x:
s(x) =
{ +1 for monomer A in position x,
−1 for monomer B in position x.
(3.1)
33002-3
L.I. Nazarov, S.K. Nechaev, M.V. Tamm
Figure 4. (Color online) The representation of two arbitrary packages in a context of 1D walks on an
adjustable substrate. Periodic regions, shown in (a) are mapped (depending on the parity) to the domains
of lamps in “off” or “on” positions, shown in (b). The white segments, shown in (c), correspond to the
domains of lamps, while red flags mark the domain walls. The human figure designates the tip of the
polymer chain, which performs a randomwalk and can push flags to the left or to the right when passing
by them.
Now, we define another Ising variable, σ(x):
σ(x) = (−1)x s(x). (3.2)
The change of variables form s(x) to σ(x) essentially simplifies the picture, because in terms of the vari-
able, σ, the adsorbing state is homogenous. The state of the chain in terms of variables σ is depicted in
figure 4 (b) by lamps switched on (if σ= 1) and off (if σ=−1). Each defect (i.e., the pair of segments A-A
or B-B) is a “domain wall” separating two states: to the left of it, the lamps are off, while to the right of it,
the lamps are on. A configuration with two defects is depicted in figure 4 (b). The defects (domain walls)
are shown in figure 4 (c) by flags. In terms of variables σ, the uniform adsorbing state corresponds to
the periodic adsorbing state of our initial Ising model, and the dynamics of folding in terms of σ means
coarsening of the Ising domains.
We can easily formulate the dynamic rules describing the evolution of the system. The tip of the
original polymer chain, shown as a human figure (“lamplighter”) in figure 4 (c) performs a walk on a
one-dimensional lattice. In the absence of any energy gain, the trans– and gauche–states are equiproba-
ble. Since the concentrations of monomers of types A and B in the chain are the same, the lamplighter
performs a symmetric random walk on a one-dimensional discrete line as it is shown in figure 4 (c).
The initial state for the spins σ is an infinite one-dimensional lattice with all domains of length 1.
Using the notations of figure 4 (c), the flags are located at each boundary between adjacent monomers,
while a single random walker (lamplighter) is initially positioned at x = 0. As the time (measured in the
number of heteropolymer monomers, N ) runs, the lamplighter performs a discrete symmetric random
walk with the following properties: i) if the lamplighter stays within a domain of lamps which all are “on”
or “off”, he does nothing with the lamps, ii) if the lamplighter crosses the domain wall, marked by a flag,
he may (with the probability 1/2) randomly shift the flag on one lattice site to the left or to the right. If two
flags meet each other at one site, they annihilate. The energy of the folding configuration is equal to the
number of flag shifts. As the flags annihilate and their concentration decreases, the energy of the folding
grows slower.
To consider the dynamics more quantitatively, we have first checked our conjectures computing nu-
merically the span, 〈R2(N )〉, of the lamplighter’s N -step walk. As shown in figure 5 (left), we have got
〈R2(N )〉 ∼ N (3.3)
as expected for the standard Brownian random walk. In terms of the original problem, this means that
the N -step random block-copolymer has a typical span of size of order
p
N on the 1D substrate.
33002-4
Lamplighter model of a copolymer adsorption
Figure 5. (Color online) (left): The average normalized span, 〈R2(N )〉, of the folded trajectory on an infinite
substrate, as a function of the number of steps, N ; (center): The normalized cumulative folding energy,
E(N )/N 3/4
, as a function of the number of steps, N ; (right): The typical normalized demixing time, τ/s3
,
of the folded trajectory as a function of the bounding box size, s.
Now, let us pay attention to the dynamics of a single flag. We have already mentioned that the lamp-
lighter passing across the flag, moves it on one lattice site to the left or to the right with equal probabilities.
This means that the flag itself performs a randomwalk with themean-square displacement 〈∆x2(s)〉 ∼ m,
wherem is the number of jumps the flag has made (i.e., the number of times the lamplighter returned to
the flag). Since the number of returns to a given point (the position of a flag),m, scales asm ∼ t 1/2
for a
one-dimensional random walk at large t , we immediately conclude that the mean-square displacement
of the flag obeys the following scaling dependence
〈∆x2(t )〉 ∼ m ∼ t 1/2. (3.4)
The waiting times between sequential visits of a given flag by the lamplighter are distributed as the
first-return times of a one-dimensional randomwalk, with a tail t−3/2
at long times. This makes themove-
ment of a single flag exactly the continuous-time random walk (CTRW) with the exponent 1/2 [3]. Note
that since the movement of different flags is governed by a single lamplighter, they are correlated. How-
ever, in the first approximation one expects that it is possible to neglect these correlations and assume
that the mean square distance, d =
√
〈(xi+1 −xi )2〉 between two neighboring flags i and i +1 located at
the points xi and xi+1, behaves as in (3.4), i.e. d 2(t ) ∼ t 1/2
.
The process of annihilation of flags is described by the equation of a one-dimensional chemical kinet-
ics X + X → 0, where the reagent X designates a flag, and X experiences a sub-diffusive random walk
at time t with the mean-square displacement 〈∆x2(t )〉 ∼ t 1/2
. Such a process has been widely studied in
chemical kinetics— see, for example, [4].
Let c(t ) be the typical concentration of X at time t within a region visited by the lamplighter. We are
interested in the survival probability of X in time, i.e., in the scaling behavior c(t ) ∼ t−α at t →∞. Each
time the two consecutive flags touch each other, they disappear. Since the flags cannot pass through each
other, the typical distance between the reagents X (flags) at time t is of the order of the explored territory
up to the time t . Therefore, c(t ) ∼ d−1(t ) or
c(t ) ∼ t−1/4
(3.5)
giving α= 1/4.
The energy of the folded chain increases by one with the probability 1/2 every time the lamplighter
visits a flagged site. There are two typical situations when it happens: a) when the lamplighter reaches
the boundary of the domain visited (recall that outside the domain visited all sites are flagged), and b)
when it encounters a flag in the bulk of the already visited region. It is easy to estimate the frequencies of
both these events. Indeed, according to (3.3), the size of the domain visited grows as t 1/2
, and the resulting
energy increment scales as
dE(t )
dt
∣∣∣∣
boundary
∼ dt 1/2
dt
∼ t−1/2. (3.6)
On the other hand, the probability for a random walk to meet a flag at time t in the domain’s bulk, scales
33002-5
L.I. Nazarov, S.K. Nechaev, M.V. Tamm
as c(t ) and, therefore,
dE(t )
dt
∣∣∣∣
bulk
∼ d−1(t ) ∼ t−1/4. (3.7)
We see that, at least on large timescales (i.e., for long chains), the second mechanism dominates, and the
energy ismostly accumulated via interaction of the lamplighter with the flags in the bulk. Integrating (3.7)
up to the total chain length, N , we end up with the scaling behavior of the average energy accumulation,
E(N ) up to the length N (N À 1):
〈E(N )〉 ∼
N∫
0
t−1/4 dt ∼ N 3/4. (3.8)
This scaling behavior is clearly seen in the numeric simulations shown in figure 5 (center), where we
have plotted the normalized cumulative energy, E(N )/N 3/4
, as a function of N . The results are obtained
by averaging over 106
random block-copolymer packaging attempts on the infinite line.
Apart from the chain adsorption on an infinite one-dimensional substrate, it seems instructive to con-
sider the case when the total size of a substrate is finite, e.g., the chain is trapped in a cage with periodic
boundary conditions. In this case (for even total number of sites, L, in the cage), a long enough chain,
once again, finally forms a state with completely regular . . . A-B-A-B . . . roof, which means, in terms of the
Ising domain representation, that a single Ising domain is formed. The natural questions to ask are as
follows: i) how long (on average) does it take for a perfect domain structure to be formed in the cage of
L sites (i.e., which is the typical “demixing time”, τ, as a function of L), and ii) what is the typical total
energy, 〈E(L)〉 (the number of defects in the structure), as a function of the cage size, L. It turns out that
the question ii) is easier to answer. Indeed, the energy E is the total number of the flag moves, regardless
of the waiting times between them. Since the moves of the flags in both directions are equiprobable, the
typical number of the moves needed for two adjacent flags to annihilate, scales as d 2
, where d is the
initial distance between them. Therefore, one expects the total accumulated energy to scale as
〈E(L)〉 ∼
N∑
n=0
d 2(n), (3.9)
where n (1 É n É L) is the number of surviving flags at a given time (all numeric constants are omitted).
Since the typical distance between the flags is d(n) ∼ L/n, we can evaluate the sum in (3.9), getting finally
〈E(L)〉 ∼ L2
(3.10)
for large L, which is perfectly confirmed by numeric simulations.
For the demixing time, we have not yet obtained the analytic scaling estimate, but according to the
numeric simulations shown in figure 5 (right), the demixing time τ(L) scales as
τ(L) ∼ L3. (3.11)
Combining (3.10) and (3.11), we get a nontrivial prediction 〈E(τ)〉 ∼ τ2/3
, different from the result of
the mean-field theory (3.8) in the infinite domain (τ plays the role of the chain length). Therefore, it
turns out that in a bounded domain, the shifts of sequential flags are highly correlated since they are
governed by a single lamplighter which cannot escape the cage. Indeed, when only several flags are left
in a cage, the correlations between their shifts become very important. A more detailed consideration of
this phenomenon will be provided elsewhere.
4. Discussion
Wehave alreadymentioned that ourmodel is reminiscent to the so-called “lamplighter randomwalk”,
widely considered in the mathematical literature (see, for example, [1, 2, 5–11]). Here is the definition
of the lamplighter graph, taken from [11]: “. . . Think of a (typically infinite) connected graph X , where
in each vertex there is a lamp that may be switched off (state 0), or switched on with q − 1 different
intensities (states 1, . . . , q −1). Initially, all lamps are turned off, and a lamplighter starts at some vertex
33002-6
Lamplighter model of a copolymer adsorption
of X and walks around. When he visits a vertex, he may switch the lamp sitting there into one of its q
different states (including “off”). Our information consists of the position x ∈ X of the lamplighter and of
the finitely supported configuration η : X →Zq = {0, . . . , q−1} of the lamps that are switched on, including
their respective intensities. The set Zq o X of all such pairs (η, x) can be in several ways equipped with a
naturally connected graph structure, giving rise to a lamplighter graph. . . ”
The key difference between the lamplighter model considered in the mathematical literature (math-
LL) and the model analyzed in our work, consists in the following: in math-LL all lamps (governed by a
single random walk) are immobile, are located at every lattice site, and do not interact with each other,
while in our model, the flags (also governed by a single random walk) can perform random local shifts
and interact (i.e., annihilate while in a contact). Despite the mentioned difference, we believe that some
important questions, such as the computation of the “mixing time” [10] on lamplighter graphs, could be
addressed in our model as well.
It seems interesting to compare the results of the local step-by-step optimization discussed at length in
this workwith the global optimization of the chain configuration. Namely, assume that after the described
packaging, the frozen configuration of folds is annealed, i.e., the existing folds are allowed to retract and
the new folds can be formed (for unchanged sequence of monomers). Clearly, the resulting ground state
energy will be lower than the one obtained in our local optimization process. The partition function of
such a system having an annealed secondary structure, apparently admits the recursive representation
similar to the dynamic programming algorithm for RNA secondary structure [12]. In the forthcoming
works we plan to provide a detailed analysis of the global optimization and its comparison with the
results of local optimization for the lamplighter random walk.
Aknowledgements
The authors are grateful to V. Avetisov, P. Krapivsky and R. Metzler for encouraging discussions. This
work was partially supported by the grants ANR-2011-BS04-013-01 WALKMAT and the IRSES project FP7-
PEOPLE-2010-IRSES 269139 DCP-PhysBio. M.T. and S.N. acknowledge the financial support of the Higher
School of Economics program for Basic Research.
References
1. Vershik A.M., In: Invariant Means on Topological Groups and Their Applications Greenleaf F.P., afterword to
Russian edn., Mir, Moscow, 1973 (in Russian).
2. Kaimanovich V.A., Vershik A.M., Ann. Probab., 1983, 11, 457–490; doi:10.1214/aop/1176993497.
3. Metzler R., Klafter J., Phys. Rep., 2000, 339, 1–77; doi:10.1016/S0370-1573(00)00070-3.
4. Yuste S.B., Lindenberg K., Chem. Phys., 2002, 284, 169–180; doi:10.1016/S0301-0104(02)00546-3.
5. Lyons R., Pemantle R., Peres Y., Ann. Probab., 1996, 24, 1993–2006; doi:10.1214/aop/1041903214.
6. Grigorchuk R.I., Zuk A., Geometriae Dedicata, 2001, 87, 209–244; doi:10.1023/A:1012061801279.
7. Kaimanovich V.A., Woess W., Ann. Probab., 2002, 30, 323–363; doi:10.1214/aop/1020107770.
8. Revelle D., Ann. Probab., 2003, 31, 1917–2300; doi:10.1214/aop/1068646371.
9. Revelle D., Electr. Commun. Probab., 2003, 8, 142–154; doi:10.1214/ECP.v8-1092.
10. Peres Y., Revelle D., Electron. J. Probab., 2004, 9, 825–845; doi:10.1214/EJP.v9-198.
11. Woess W., Comb. Probab. Comput., 2005, 14, 415–433; doi:10.1017/S0963548304006443.
12. Nechaev S.K., TammM.V., Valba O.V., J. Phys. A: Math. Theor., 2011, 44, 195001;
doi:10.1088/1751-8113/44/19/195001.
33002-7
http://dx.doi.org/10.1214/aop/1176993497
http://dx.doi.org/10.1016/S0370-1573(00)00070-3
http://dx.doi.org/10.1016/S0301-0104(02)00546-3
http://dx.doi.org/10.1214/aop/1041903214
http://dx.doi.org/10.1023/A:1012061801279
http://dx.doi.org/10.1214/aop/1020107770
http://dx.doi.org/10.1214/aop/1068646371
http://dx.doi.org/10.1214/ECP.v8-1092
http://dx.doi.org/10.1214/EJP.v9-198
http://dx.doi.org/10.1017/S0963548304006443
http://dx.doi.org/10.1088/1751-8113/44/19/195001
L.I. Nazarov, S.K. Nechaev, M.V. Tamm
Модель лiхтарника адсорбцiї випадкового кополiмера на
лiнiї
Л.I. Назаров1, С.К. Нєчаєв2,3,4,М.В. Тамм1,4
1 Фiзичний факультет,Московський державний унiверситет iм.М.В. Ломоносова,Москва, РФ
2 Унiверситет Парi-Сюд/CNRS, Орсе, Францiя
3 Фiзичний iнститут iм. П.Н. Лєбєдєва, РАН,Москва, РФ
4 Факультет прикладної математики, Нацiональний дослiдницький унiверситет “Вища школа економiки”,
Москва, РФ
Ми представляємо модель послiдовного самопакування AB-диблокового випадкового кополiмера з ло-
кальними замороженими взаємодiями на одновимiрнiй нескiнченiй липкiй основi. Припускається, що
контакти A-A i B-B є сприятливi, тодi як A-B є несприятливими. Положення нового мономера, що дода-
ється, виберається з точки зору мiнiмiзацiї енергiї локального контакту. Модель демонструє саморганi-
зовану поведiнку з нетривiальною залежнiстю загальної енергiї, E (числа несприятливих контактiв), вiд
числа мономерiв ланцюга, N : E ∼ N 3/4 для замороженого хаотичного рiвноймовiрного розподiлу A-
i B-мономерiв вздовж ланцюга. Модель розглядається шляхом зiставлення її з випадковим блуканням
лiхтарника i дифузiйно-контрольованою хiмiчною реакцiєю типу X + X → 0 з субдифузiйним рухом ре-
агентiв.
Ключовi слова: локальна оптимiзiцiя, згортання гетерополiмера, випадкове блукання лiхтарника,
субдифузiйна хiмiчна реакцiя
33002-8
Introduction
The model
Results
Discussion
|