Master equation approach to protein folding
The dynamics of two 12-monomer heteropolymers on the square lattice is studied exactly within the master equation approach. The time evolution of the occupancy of the native state is determined. At low temperatures, the median folding time follows the Arrhenius law and is governed by the longest...
Gespeichert in:
| Veröffentlicht in: | Condensed Matter Physics |
|---|---|
| Datum: | 1999 |
| Hauptverfasser: | , , |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
1999
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/120399 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Master equation approach to protein folding / M. Cieplak, M. Henkel, J.R. Banavar // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 369-378. — Бібліогр.: 17 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-120399 |
|---|---|
| record_format |
dspace |
| spelling |
Cieplak, M. Henkel, M. Banavar, J.R. 2017-06-12T06:53:00Z 2017-06-12T06:53:00Z 1999 Master equation approach to protein folding / M. Cieplak, M. Henkel, J.R. Banavar // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 369-378. — Бібліогр.: 17 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.369 PACS: 87.15.By, 87.10.+e https://nasplib.isofts.kiev.ua/handle/123456789/120399 The dynamics of two 12-monomer heteropolymers on the square lattice is studied exactly within the master equation approach. The time evolution of the occupancy of the native state is determined. At low temperatures, the median folding time follows the Arrhenius law and is governed by the longest relaxation time. For both good and bad folders, significant kinetic traps appear in the folding funnel and the kinetics of the two kinds of folders are quite similar. What distinguishes between the good and bad folders are the differences in their thermodynamic stabilities. За допомогою методу керуючого рівняння точно проаналізована динаміка двох 12-мономерних гетерополімерів на квадратній гратці. Визначена часова еволюція зайнятості нативного стану. При низьких температурах середній час скручування підлягає закону Ареніуса і визначається найдовшим часом релаксації. Для білків, що добре скручуються, з’являються суттєві кінетичні пастки у наборі послідовних конформацій, у той час як для білків, що погано скручуються, пастки присутні також і в ділянках, що не відповідають нативній конформації. We thank Jan Karbowski for collaboration and T. X. Hoang for discussions. This work was supported by KBN (Grant No. 2P03B-025-13), Polonium, CNRS-UMR 7556, NASA, and the Applied Research Laboratory at Penn State. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Master equation approach to protein folding Метод керуючого рівняння у явищі скручування білків published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Master equation approach to protein folding |
| spellingShingle |
Master equation approach to protein folding Cieplak, M. Henkel, M. Banavar, J.R. |
| title_short |
Master equation approach to protein folding |
| title_full |
Master equation approach to protein folding |
| title_fullStr |
Master equation approach to protein folding |
| title_full_unstemmed |
Master equation approach to protein folding |
| title_sort |
master equation approach to protein folding |
| author |
Cieplak, M. Henkel, M. Banavar, J.R. |
| author_facet |
Cieplak, M. Henkel, M. Banavar, J.R. |
| publishDate |
1999 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| title_alt |
Метод керуючого рівняння у явищі скручування білків |
| description |
The dynamics of two 12-monomer heteropolymers on the square lattice is
studied exactly within the master equation approach. The time evolution
of the occupancy of the native state is determined. At low temperatures,
the median folding time follows the Arrhenius law and is governed by the
longest relaxation time. For both good and bad folders, significant kinetic
traps appear in the folding funnel and the kinetics of the two kinds of folders
are quite similar. What distinguishes between the good and bad folders are
the differences in their thermodynamic stabilities.
За допомогою методу керуючого рівняння точно проаналізована динаміка двох 12-мономерних гетерополімерів на квадратній гратці.
Визначена часова еволюція зайнятості нативного стану. При низьких температурах середній час скручування підлягає закону Ареніуса і визначається найдовшим часом релаксації. Для білків, що добре скручуються, з’являються суттєві кінетичні пастки у наборі послідовних конформацій, у той час як для білків, що погано скручуються,
пастки присутні також і в ділянках, що не відповідають нативній конформації.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120399 |
| citation_txt |
Master equation approach to protein folding / M. Cieplak, M. Henkel, J.R. Banavar // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 369-378. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT cieplakm masterequationapproachtoproteinfolding AT henkelm masterequationapproachtoproteinfolding AT banavarjr masterequationapproachtoproteinfolding AT cieplakm metodkeruûčogorívnânnâuâviŝískručuvannâbílkív AT henkelm metodkeruûčogorívnânnâuâviŝískručuvannâbílkív AT banavarjr metodkeruûčogorívnânnâuâviŝískručuvannâbílkív |
| first_indexed |
2025-11-26T01:39:30Z |
| last_indexed |
2025-11-26T01:39:30Z |
| _version_ |
1850603179677319168 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 369–378
Master equation approach
to protein folding
M.Cieplak 1 , M.Henkel 2 , J.R.Banavar 3
1 Institute of Physics, Polish Academy of Sciences, and College of
Sciences, 02-668 Warsaw, Poland
2 Laboratoire de Physique des Matériaux, Université Henri Poincaré
Nancy I, F-54506 Vandœuvre, France
3 Department of Physics and Center for Materials Physics, 104 Davey
Laboratory, The Pennsylvania State University, University Park, PA 16802
Received June 25, 1998
The dynamics of two 12-monomer heteropolymers on the square lattice is
studied exactly within the master equation approach. The time evolution
of the occupancy of the native state is determined. At low temperatures,
the median folding time follows the Arrhenius law and is governed by the
longest relaxation time. For both good and bad folders, significant kinetic
traps appear in the folding funnel and the kinetics of the two kinds of folders
are quite similar. What distinguishes between the good and bad folders are
the differences in their thermodynamic stabilities.
Key words: protein folding, kinetic equations
PACS: 87.15.By, 87.10.+e
1. Introduction
A denatured protein folds into a compact native state in a time of the order
of a millisecond after the physiological conditions are restored [1]. This process is
reversible and the native state is believed to coincide with the ground state of the
system. Studies of simplified lattice models of proteins have provided many insights
into the dynamics of the folding process [2]. The crucial feature that makes lattice
models useful is the possibility of enumerating the complete set of conformations and
determining which of them is the ground state. This can be accomplished provided
the length of the heteropolymer, N , is small. Such studies have elucidated the role
of thermodynamic stability [2,3], stability against mutations [4] and the existence of
a linkage between the rapid folding and the stability of the native state [3].
The key concept that has been introduced to explain the rapid folding occurring
in natural proteins is that of the folding funnel [5,6] – a set of conformations that are
c© M.Cieplak, M.Henkel, J.R.Banavar 369
M.Cieplak, M.Henkel, J.R.Banavar
smoothly connected to the native state, as indicated schematically at the top of figure
1. It is expected that for random sequences of aminoacids there exist competing
basins of attraction which would otherwise trap the system away from the native
state. This corresponds schematically to what is shown at the bottom of figure 1.
It isn’t easy to identify the states belonging to the funnel because of an enormous
number of conformations present even for a small N as well as because the problem
is a dynamical one. Possible approaches include the monitoring of frequencies of
passages between various states in Monte Carlo trajectories [6] or the mapping of
states into underlying valleys of effective states [7,8].
Figure 1. A schematic representation of the
phase space properties of good (top) and
bad (bottom) folders. The vertical axis cor-
responds to energy and the horizontal axis
to a “coordinate” in the phase space.
Essentially all approaches to the
studies of the folding dynamics are
restricted to Monte Carlo simula-
tions that start from a few randomly
chosen initial conformations [9]. The
only exception is an approach due to
Chan and Dill [10] in which an enu-
meration of transition rates between
classes of conformations which have
the same number of contacts and are
a given number of kinetic steps away
from the native state.
Recently, we presented an exact
method to study the dynamics of
short model proteins [11] which was
based on the master equation [13].
We have specifically considered two
N=12 sequences, called A and B,
which are placed on a square lattice.
The dynamics of these sequences can
be studied exactly because the se-
quences can acquire only N=15037
conformations. The two sequences
have the same set of contact energies Bij but their assignment to various monomers
i and j is different with the result that A is a good folder and B is a bad folder.
The basic finding of [10] is that the qualitative picture corresponding to figure 1
is absolutely correct. By identifying kinetic trap states responsible for the slowest
dynamical processes in the system we could demonstrate that the traps for sequence
A are within the folding funnel whereas for sequence B the relevant traps form a
valley which competes with the native valley similar to the bottom of figure 1. In
the current paper we provide a deeper characterization and comparisons of the two
sequences and elucidate the nature of the trap states.
The sequences that we study are described by the Hamiltonian
H =
∑
ij
Bij∆ij , (1)
370
Master equation approach to protein folding
Figure 2. Top: The native conformation and
its energy for sequence A. The enlarged
circle shows the first monomer. Main: Dy-
namical data for the folding. The solid line
marked by tfold gives the median folding
time derived from 1000 Monte Carlo trajec-
tories. The solid line τ1 is the longest re-
laxation time. The broken line t 1
2
with the
black circles gives the time for P0(t) to reach
1
2P0 from the uniformly random initial state.
The dotted line tA is a fit of the Monte Car-
lo data to the Arrhenius law with δE=2.76.
The arrow at the top indicates the value of
the folding transition temperature.
where ∆ij indicates the contact in-
teraction Bij assigned to monomers
which are geometrical nearest neigh-
bours on the lattice but are not
neighbours along the sequence. In
such arrangements ∆ij is equal to 1,
otherwise it is 0. The values of the
25 couplings are chosen as Gaussian
numbers which are centred around
-1 to provide an overall attraction,
as detailed in [10]. The ground state
of sequence A is maximally compact
and it fills the 3×4 lattice, as shown
at the top of figure 2. For sequence
B, the ground state, shown at the
top of figure 3, is doubly degener-
ate. Both of the ground states are
compact. However they are not max-
imally compact. They differ merely
by a placement of one end monomer
and therefore they were considered
as an effective single native state.
We have found that the dynam-
ics of A and B are superficially sim-
ilar: for both, the median folding
time, tfold, and the longest relaxation
time, τ1 diverge at low T according
to an Arrhenius law. The temper-
ature Tmin at which folding to the
native state proceeds the fastest is
about the same for both sequences.
It is the location of the folding transition temperature, T f , with respect to Tmin which
distinguishes between sequences A and B. Tf is defined as the temperature at which
the equilibrium value of the probability to occupy the native state, P0, crosses
1
2
and is a measure of thermodynamic stability. For bad folders, T f is well below Tmin
and thus a substantial occupation probability for the native state is found only in a
temperature range in which the dynamics are glassy. For sequence A, the values of T f
and Tmin are 0.71 and 1.0±0.1, respectively, while for sequence B, the corresponding
values are 0.01 and 1.1 ± 0.1. Thus the two sequences are dynamically similar but
the equilibrium properties differ dramatically.
371
M.Cieplak, M.Henkel, J.R.Banavar
2. Methods
Figure 3. Same as figure 2, but for sequence
B. For the curve tA, δE=3.55. There are two
native conformations with the same energy.
We study our sequences through
an analysis of the master equation
and then compare the results to
those obtained by the Monte Carlo
approach. The master equation does
not deal with a specific trajectory
but with an ensemble of trajectories
and it reads
∂Pα
∂t
=
∑
β 6=α
[
w(β → α)Pβ (2)
−w(α → β)Pα
]
,
where Pα = Pα(t) is the probabil-
ity of finding the sequence in con-
formation α at time t. The quantity
wαβ = w(β → α) is the transition
rate from conformation β to confor-
mation α. Of course, writing a mas-
ter equation relies on the assumption
that a Markov chain description ad-
equately describes protein kinetics. In particular, memory effects are assumed to be
negligible. The actual derivation of a master equation remains a formidable problem
in itself [12].
One may bring this into a matrix form by letting ~P = (P1, . . . , PN ) and
mαβ = −wαβ 6 0, α 6= β; , mαα =
∑
β 6=α
wβα . (3)
The master equation then takes the form of an imaginary-time Schrödinger equation
[13,14]
∂t ~P = −M̂ ~P , (4)
where the mαβ are the matrix elements of M̂ .
The time-dependence state vector ~P (t) at time t = nτ0 can be obtained by
applying n times the recursion ~P ((n + 1)τ0) = (1− τ0M̂)~P (nτ0) to ~Pin, where τ0 is
a microscopic time associated with a single move and ~Pin is some initial probability
distribution. On the other hand, the spectrum of relaxation times τλ = 1/(Re Mλ) >
0 follows directly from the eigenvalues Mλ of M̂ .
The transition rates wαβ are designed so that a stationary solution of the master
equation corresponds to equilibrium with the Hamiltonian given by equation (1).
This can be accomplished provided the detailed balance condition
wαβP
eq
β = wβαP
eq
α (5)
372
Master equation approach to protein folding
is satisfied. Here, P eq
α ∼ e−Eα/T is a stationary solution of the master equation and
Eα is the energy of the sequence in conformation α. Equation (5) is satisfied by
any wαβ = fαβ exp [−(Eα − Eβ)/2T ] provided only that fαβ = fβα. Here, we choose
wαβ = w
(1)
αβ + w
(2)
αβ , where
w
(σ)
αβ =
2
τ0
Rσ cosh [(Eα − Eβ) /T ] (6)
with R1+R2 = 1. Here, σ = 1 and 2 refer to the single- and double-monomer moves
respectively. It is understood that w
(σ)
αβ = 0 if there is no move of type σ linking
β with α. The choice (6) guarantees that transition rates are finite and bounded
for all temperatures. Similar to [3], we focus on R1 = 0.2 and take the single and
two-monomer (crankshaft) moves as in [10].
Because of the detailed balance condition, the eigenvalues Eα are not calculated
by diagonalizing M̂ directly, but by diagonalizing an auxiliary matrix K̂ with ele-
ments
kαβ =
vβ
vα
mαβ = kβα , (7)
where vα = e−Eα/2T . The master equation can be then rewritten as
∂t ~Q = −K̂ ~Q , (8)
where K̂ is real symmetric and Qα = Pα/vα. This equation is then diagonalized
using the standard symmetric Lanczos algorithm without reorthogonalization, as
described in [15,16] and then the eigenmodes of M̂ are determined. In particular,
since K̂ and M̂ are related by a similarity transformation, all eigenmodes Mλ are
real.
The eigenvector corresponding to Mλ = 0, i.e. to the infinite relaxation time, de-
termines the equilibrium occupancies of the conformations. This eigenvector needs to
be normalized to satisfy its probabilistic interpretation. The longest finite relaxation
time τ1 = 1/M1 is found from the smallest non-zero eigenvalue M1.
Our Monte Carlo simulations were done in a way that satisfies the detailed
balance conditions and were devised along the lines described in [10]. For poly-
mers, getting these conditions satisfied is not trivial because each conformation has
its own number, K, of allowed moves that the conformation can make. Thus the
propensities to make a move in a unit time vary from conformation to conforma-
tion and the effective “activities” of the conformations need to be matched. This
can be accomplished by first determining the maximum value of K, Kmax. For the
12-monomer chain on the square lattice, Kmax is equal to 14. We associate a single
time unit with the conformations in which K=Kmax. This means that each move
allowed is attempted always with probability 1/Kmax. For a conformation with K
allowed moves, the probability to attempt any move is then K/Kmax and the proba-
bility not to carry out any attempt is 1 − K/Kmax. The attempted moves are then
accepted or rejected as in the standard Metropolis procedure. The probability of a
single monomer move (an end flip or a switch in a pair of bonds that make the 90◦
angle) is additionally reduced by the factor of 0.2 and an allowed double monomer
373
M.Cieplak, M.Henkel, J.R.Banavar
crankshaft move by 0.8. The time is then measured in terms of the total number of
the Monte Carlo attempts divided by Kmax. This scheme not only establishes the
detailed balance conditions but it also uses less CPU compared to a process in which
moves are attempted regardless of their being allowed or not allowed.
3. Results
Figure 4 shows the 4 longest finite relaxation times for sequence A as a function
of temperature. It is seen that they are all roughly proportional to each other and
they all appear to diverge at T=0, albeit with different energy barriers. The longest
relaxation time follows the Arrhenius law, τ1 ∼ eδE1/T with δE1 of about 4.1. The
Monte Carlo derived median folding time also follows the Arrhenius law at low tem-
peratures but with δE = 2.76± 0.06. The overall Arrhenius-like behaviour suggests
that the physics of folding at low T ’s is dominated by processes that establish equi-
librium. Their longest time scale is then set by τ1. At temperatures above Tmin, tfold
no longer follows τ1. Here, the physics of folding is dominated by the statistics of rare
events: the system establishes equilibrium rapidly and then folding is reached by a
search in equilibrium. The search becomes more and more random when T becomes
larger and larger. For sequence B, similar results are obtained but the Arrhenius
barrier in tfold is 3.55± 0.06, whereas δE1 = 4.0±0.06. Figures 2 and 3 show fits to
the Arrhenius law for sequences A and B respectively.
Figure 4. The four longest relaxation times
versus T for sequence A. The very longest,
τ1, is drawn by a solid line whereas the re-
maining three – by the dotted lines. The
value of Tf is indicated by an arrow and the
Monte Carlo data on tfold are repeated from
figure 2 for a comparison.
We now focus on the time evo-
lution of the probability, P0(t), to
occupy the native state, where t de-
notes time. This probability depends
on the initial conditions. The solid
lines in figure 5 show the evolution
of P0(t) when in the initial state all
conformations have equal probabil-
ity of 1/N to be occupied. The main
figure is for sequence A and the time
evolution is shown for three temper-
atures: 1.2, 0.9 and 0.6. The data
for the lowest of these temperatures
are also compared with the Monte
Carlo evolution. The Monte Car-
lo data, obtained by starting from
random conformation, agree with
the exact evolution. The time scales
at which the equilibrium saturation
values of P0 are reached are of order
τ1. The top portion of figure 5 shows
P0(t) for sequence B. It is seen that
the saturation levels are significantly
374
Master equation approach to protein folding
lower for B than for A at the same temperatures. This reflects a significantly lower
value of Tf for B. We also observe, by looking at figures 2 and 3 that the median
folding time at low temperatures appears to coincide with the time needed for P 0(t)
to reach half of its equilibrium value.
Figure 5. Probability of occupation of the
native state, P0(t), of sequence A, for three
values of the temperatures, indicated on the
right. P0(∞) agrees with the equilibrium
value. The solid lines correspond to the uni-
formly random initial condition and the dot-
ted lines correspond to an initial placement
of the system in the trap state. The squares
correspond to Monte Carlo results, based on
200 random starting conformations. The top
figure shows P0(t) for sequence B. The ini-
tial condition is that of uniform randomness.
The y-axis scale is indicated on the right.
The dotted line in figure 5 cor-
responds to the initial state being
the kinetic “trap” conformation [17]
which is the strongest obstacle in
reaching equilibrium. If the system
starts in the trap state, it will still
reach the final equilibrium in a time
scale set by τ1 but at shorter times
the occupancy of the native state is
significantly smaller compared to the
start from a totally random state.
In general, it is not easy to de-
termine which conformation forms
the most potent trap. One may at-
tempt to determine it by monitor-
ing the occupation of the local en-
ergy minima in a Monte Carlo pro-
cess performed at very low tempera-
tures. This approach has been suc-
cessfully applied in [8]. Here, the
trap state is determined in an ex-
act way by studying the eigenvector
corresponding to the longest relax-
ation time and by identifying the lo-
cal energy minima which contribute
the most to this eigenvector at low
T . The largest weight is associated
with the native state. Usually, the
second largest weight corresponds to
the most relevant trap. This happens provided the second largest occupancy is in
a local energy minimum state. A non-minimum state, immediately preceding the
native state kinetically, may also possess a significant occupancy but it does not
constitute a trap. Thus the search for the most relevant trap goes after non-native
local energy minima which have the biggest occupancy. In the limit of T → 0,
weights associated with all other local energy minima states become small.
We now focus on the interpretation of the trap states to demonstrate the qual-
itative validity of figure 1. We place the system in the trap state and ask what is
the best trajectory which leads from the trap state to the native state. The best
trajectory is defined to be one in which the highest energy state is lower than the
highest energy state on all other trajectories.
375
M.Cieplak, M.Henkel, J.R.Banavar
Figure 6. The best trajectory linking the
trap state of sequence A (shown at the top
left) and the native state. The energies in-
volved are shown below each conformation.
The biggest single step and global energy
expenses are indicated at the bottom.
Figure 7. As in figure 6 but for sequence B.
Figure 6 shows the trap state
(of energy -8.9627) for sequence A
and the best corresponding trajec-
tory that connects it to the native
state. The trajectory reaches the na-
tive state in 8 steps and it elevates
in energy by 4.5323, relative to the
the energy of the trap state, after ac-
complishing the second step. Thus it
takes only two uphill steps to enable
a flow to the native state. The first
of these steps breaks two contacts
and it costs 2.8823 in energy. Thus
this biggest single step energy cost is
what determines the value of the en-
ergy barrier in the Arrhenius law for
tfold, as derived through the Monte
Carlo technique. On the other hand,
δE1, that determines τ1, appears to
be governed by the energy costs to
make the initial two-step climb to
the point of the highest elevation. In
a Monte Carlo process, two consec-
utive climbs up are not very likely
at low T ’s and can be missed. How-
ever, the exactly derived longest re-
laxation time indicates the true na-
ture of the barrier.
The kinetics of sequence B is
quite similar to that of sequence
A. Thus what distinguishes the two
sequences are their thermodynamic
stabilities as measured by Tf . Figure
7 shows the steps of the best tra-
jectory from the the most important
trap state for sequence B (of energy –
8.4431) to the native state. The trap
state is almost identical in shape to
the two native conformations: it is
only the positioning of the middle
portion of the conformation that is
not quite right. And yet it takes 10 steps, like in figure 6, to accomplish the transi-
tion between the trap and native states. The overall barrier that is climbed is equal
to 4.4 which is close to the Arrhenius barrier δE1 in τ1 of 4.0. The biggest single step
376
Master equation approach to protein folding
energy cost of 2.7478 on the trajectory is, however, further away from δE in tfold of
about 3.6. The important observation is that for the bad folder B the most relevant
trap conformation is still in the folding funnel of the native state. The presence of
other low lying valleys that compete with the native valley does not really affect the
trapping effects in the native funnel. Instead it generates low stability of the native
state through a substantial equilibrium probability of being in some other valleys.
Even though our exact results have been obtained in a toy model of only 12
monomers, it is expected that the essential physics of protein folding and the dis-
tinction between the good and bad folders are as illustrated here. Tools to study
larger and off-lattice systems in this spirit remain to be developed.
We thank Jan Karbowski for collaboration and T. X. Hoang for discussions. This
work was supported by KBN (Grant No. 2P03B-025-13), Polonium, CNRS-UMR
7556, NASA, and the Applied Research Laboratory at Penn State.
References
1. Creighton T.E. Proteins. Freeman, New York, 1993.
2. Wolynes P.G., Onuchic J.N., Thirumalai D. Navigating the folding routes. // Sci-
ence, 1995, vol. 267, p. 1619; Dill K.A., Bromberg S., Yue S., Fiebig K., Yee K.M.,
Thomas D.P., Chan H.S. Principles of protein folding – A perspective from simple
exact models. // Protein Sci. 1995, vol. 4, p. 561; Camacho C.J., Thirumalai D. Ki-
netics and thermodynamics of folding in model proteins. // Proc. Nat. Acad. Sci.
USA, 1993, vol. 90, p. 6369; Chan H.S. and Dill K.A. The protein folding problem. //
Phys. Today, 1993 (February), vol. 46, p. 24.
3. Sali A., Shakhnovich E., Karplus M. How does a protein fold? // Nature, 1994, vol. 369,
p. 248.
4. Li H., Helling R., Tang C., Wingreen N. Emergence of preferred structures in a simple
model of protein folding. // Science, 1996, vol. 273, p. 666; Vendruscolo M., Mari-
tan A., Banavar J.R. Stability threshold as a selection principle for protein design. //
Phys. Rev. Lett. 1997, vol. 78, p. 3967.
5. Onuchic J.N., Wolynes P.G., Luthey-Schulten Z. Towards an outline of the topog-
raphy of a realistic protein-folding funnel. // Proc. Natl. Acad. Sci., 1995, vol. 92,
p. 3626; Bryngelson J.D., Onuchic J.N., Socci N.D., Wolynes P.G. Funnels, pathways
and the energy landscape of protein folding: A synthesis. // Proteins: Struct. Funct.
and Genet., 1995, vol. 21, p. 167.
6. Leopold P.E., Montal M., Onuchic J.N. Protein folding funnels: a kinetic approach
to the sequence-structure relationship. // Proc. Natl. Acad. Sci. USA, 1992, vol. 89,
p. 8721.
7. Cieplak M., Vishveshwara S., Banavar J.R. Cell dynamics of model proteins. // Phys.
Rev. Lett., 3681 vol. 77, p. 3681; Cieplak M., Banavar J.R. Cell dynamics of folding
in two dimensional model proteins. // Fold. Des., 1997, vol. 2, p. 235.
8. Cieplak M., Hoang T.X. Coarse grained description of the protein folding. // Phys.
Rev. E, 1998, vol. 58, p. 3589.
9. Socci N.D., Onuchic J.N. Folding kinetics of proteinlike heteropolymers. // J. Chem.
Phys., 1994, vol. 101, p. 1519.
377
M.Cieplak, M.Henkel, J.R.Banavar
10. Chan H.S., Dill K.A. Energy landscape and the collapse dynamics of homopolymers. //
J. Chem. Phys. 1993, vol. 99, p. 2116; Transition states and folding dynamics of
proteins and heteropolymers. // J. Chem. Phys., 1994, vol. 100, p. 9238.
11. Cieplak M., Henkel M., Karbowski J., Banavar J.R. Kinetic traps in exactly solvable
models of proteins. // Phys. Rev. Lett., 1998, vol. 80, p. 3654.
12. Kreuzer H.J. Nonequilibrium Thermodynamics and its Statistical Foundations. Ox-
ford, Oxford University Press, 1981, ch. 10.
13. van Kampen N.G. Stochastic Processes in Physics and Chemistry. Amsterdam, North
Holland, 1981; Schnakenberg J. Network theory of microscopic and macroscopic be-
haviour of master equation systems. // Rev. Mod. Phys., 1976, vol. 48, p. 571.
14. Alcaraz F.C., Droz M., Henkel M., Rittenberg V. Reaction-diffusion processes, critical
dynamics and quantum chains. // Ann. of Phys., 1994, vol. 230, p. 250.
15. Henkel M. Conformal Invariance and Critical Phenomena. Heidelberg, Springer, 1999,
ch. 9.
16. Dagotto E., Correlated electrons in hig-temperature superconductors. // Rev. Mod.
Phys., 1994, vol. 66, p. 763
17. Mirny L.A., Abkevich V., Shakhnovich E.I. Universality and diversity of the protein
folding scenarios: A comprehensible analysis with the aid of a lattice model. // Fold.
Des., 1996, vol. 1, p.103.
Метод керуючого рівняння
у явищі скручування білків
М.Цєпляк 1 , М.Генкел 2 , Дж.Р.Беневер 3
1 Інститут фізики Польської Академії Наук та Коледж природничих
наук, 02–668 Варшава, Польща
2 Лабораторія фізики матеріалів, Університет Генрі Пуанкаре Нансі I,
F–54506 Вандоувр, Франція
3 Факультет фізики і Центр фізики матеріалів, лабораторія Дейвей
104, університет штату Пенсильванія, Юнівесіті Пак, PA 16802, США
Отримано 25 червня 1998 р.
За допомогою методу керуючого рівняння точно проаналізована ди-
наміка двох 12-мономерних гетерополімерів на квадратній гратці.
Визначена часова еволюція зайнятості нативного стану. При низь-
ких температурах середній час скручування підлягає закону Ареніу-
са і визначається найдовшим часом релаксації. Для білків, що до-
бре скручуються, з’являються суттєві кінетичні пастки у наборі послі-
довних конформацій, у той час як для білків, що погано скручуються,
пастки присутні також і в ділянках, що не відповідають нативній кон-
формації.
Ключові слова: скручування білків, кінетичні рівняння
PACS: 87.15.By, 87.10.+e
378
|