Supporting information
Essential function of the N-termini tails of the proteasome for the gating
mechanism revealed by molecular dynamics simulations
Hisashi Ishida
Simulated Annealing in vacuum
Hereafter, all the MD simulations were carried out using an MD simulation program
called SCUBA 1 2 3 with the AMBER parm99SB force-field 4.
In order to optimize the
conformation of the N-termini tails, simulated annealing (SA) was performed in
vacuum by assuming the distance-dependent dielectric constant of 4.0r with the value of
r in Angstrom.
Non-bonded interactions were evaluated with a cut-off radius of 12 Å.
The time-step of 0.5 fs was used throughout the SA. As the surfaces of the X-ray
structures were removed for simplicity, the ordered- and disordered-gate models were
structurally fragile. To fix the structures, the majority of the atoms in the models were
restrained; only the atoms of the first seven (ten) residues of the N-termini tails were
1
free to move for the ordered (disordered)-gated models respectively, and other heavy
atoms were restrained by a force constant of 10 kcal/mol/Å2 in the SA simulation. For
each system, two types of simulations were carried out; one is a simulation in which the
substrate was located in the antechamber at (r, z) = (0 Å, -20 Å), and the other is a
simulation in which the substrate was located in the activator at (r, z) = (0 Å, 35 Å).
The center of mass of the substrate was restrained by a force constant of
1.0 kcal/mol/Å2.
The system was heated from 0 to 1200 K during the first 100 ps and was then
equilibrated for 400 ps.
The equilibrated system was then gradually cooled for 500 ps
from 1200 K to 300 K.
The SA was repeated 10 times and the resulting coordinate
sets were stored as possible conformations of the missing N-termini tails at local
minimum energy regions.
Each of the 10 conformations was minimized for 500 steps
using the steepest descent algorithm followed by 5,000 steps of the conjugate gradient
algorithm without constraining the missing N-termini tails. Then, as the representative
of the 10 conformations, the minimum energy (energy was defined as the total of the
internal energy of the N-termini tails and the interaction energy between the N-termini
tails and the other part of the complex) was selected.
2
Model of proteasome – activator – substrate complex in water
After the SA, the four models were placed in an aqueous medium.
The PAN-like –
CPO/D and PA26 – CPO/D complexes were each placed in a rectangular box (96 Å × 92
Å × 122 Å), (97 Å × 93 Å ×124 Å), (89 Å × 90 Å × 114 Å) and (91 Å × 90 Å × 116 Å),
respectively.
In this box, all the atoms of the models were separated more than 6 Å
from the edge of the box.
The rather short distance of 6 Å was used as the dynamics
of the substrate and the N-termini tails were the main focus of this study.
The
dynamics of the water molecules outside the fixed surface of the models was assumed
not to have a significant influence on the dynamics of the substrate and the N-termini
tails.
To neutralize the charges of the models, sodium ions were placed at positions
with large negative electrostatic potential.
Moreover, sodium and chloride ions were
added to the box at a concentration of 100 mM NaCl.
were added to surround the four models.
Then TIP3P water molecules
5
Consequently, the four systems, the
PAN-like – CPO/D and PA26 – CPO/D complexes, comprised 106,028 atoms (the
complex – 27,601 atoms, substrate – 145 atoms, 26,044 water molecules, 60 sodium
and 90 chloride ions), 111,410 atoms (the complex – 27,468 atoms, substrate – 145
atoms, 27,880 water molecules, 60 sodium and 97 chloride ions), 92,182 atoms (the
complex – 27,398 atoms, substrate – 145 atoms, 21,501 water molecules, 60 sodium
3
and
76
chloride
ions),
and
95,191
atoms
(the
complex – 27,265
atoms,
substrate – 145 atoms, 22,546 water molecules, 60 sodium and 83 chloride ions),
respectively.
Energy minimization of each system was carried out to alleviate unfavorable
interactions between the complex and the water molecules using the steepest descent
algorithm for 500 steps and following the conjugate gradient algorithm for 5,000 steps.
Harmonic restraints with a force constraint of 10.0 kcal/mol/Å2 were applied to all the
heavy atoms of the molecules.
The dielectric constant used was 1.0 and the van der
Waals interactions were evaluated with a cut-off radius of 8 Å. The particle-particle
particle-mesh (PPPM) method
6
was used for the electrostatic interactions.
For the
PPPM calculations, charge grid sizes of (90 Å × 90 Å × 120 Å) were chosen for the
PAN-like – CPO/D complex, and (96 Å × 90 Å ×120 Å) and (96 Å × 96 Å × 125 Å) were
chosen for the PA26 – CPO/D complex, respectively, to set charge grid spacing close to
1 Å.
The charge grid was interpolated using a spline of the order of seven, while the
force was evaluated using a differential operator of the order of six 6.
Free-energy calculation by ABMD implemented in SCUBA
To evaluate the free-energy of the translocation of the peptide inside the complex, the
4
adaptively biased molecular dynamics (ABMD) method
exchange method 8 was employed in SCUBA.
7
combined with replica
The reaction coordinate of z was set at
the D7 symmetry axis of the CP (see Fig. 1). The z-component of the center of mass
of the substrate was employed as the value of the reaction coordinate.
The equations
of motion used in the ABMD method are expressed as 7:
ma
d 2ra

 Fa 
U t |   R   ,
2
dt
ra 
(S1)
U (t | z ) k BT

K  z    R   ,
t
F
where R  r1,..., rN  is the coordinates of the substrate, and N is the number of atoms of
the substrate (N = 145).   R  is a function to give the value of the reaction coordinate
(z-component of the center of mass of the substrate).
kB is the Boltzmann constant, T
is the constant temperature,F is the flooding time scale, and K is the kernel which has
distribution around the reaction coordinate.
The first equation is for atom a, with an
additional force coming from the biasing potential U(t|z) with an ordinary atomic force
of Fa. The second equation is the time-evolving equation of the biasing potential.
For large enough F and small enough width of the kernel, U(t|z) converges towards the
free-energy F(z) times -1 as t →∞ 7.
In the ABMD, the range of the reaction
coordinate from zmin to zmax is discretized to n grids with an interval of z such as
zmin = z0, z1, …, zn-1 = zmax. The biasing potential is expanded in terms of a third order
5
B-spline function 7 as follows:
U (t | z ) 
n
U
k 1
k
 z  zmin
(t ) Bk 
 z

,

(S2)
( z  i )2 (| z  i | 2) / 2  2 / 3,

Bi ( z )  (2 | z  i |)3 / 6,
0,

0 | z  i | 1,
(S3)
1 | z  i | 2,
otherwise.
Uk(t) is the coefficient of Bk at the discrete reaction coordinates. The third B-spline
Bi(z) has more than zero in the range of | z – i | < 2 with a center of z = i.
B-spline in the literature of Babin
7
(B() of the
is equivalent to Bi(x) by setting z - i.)
Therefore, the number of Uk(t) which is required to determine U(t|z) is n+2; in addition
to the coefficients Uk(t), k = 0, …, n-1, two coefficients of U-1(t) and Un(t) are required.
At the implementation of the ABMD method in SCUBA, the force derived from the
biasing potential was assumed to be zero at z = zmin and z = zmax,
U (t | z )
z

z  zmin
U (t | z )
z
 0,
(S4)
z  zmax
so that U(t|z) = U(t|zmin) at z ≤ zmin and U(t|z) = U(t|zmax) at z ≥ zmax.
This boundary
condition is satisfied when
U 1 (t )  U1 (t ), U n (t )  U n2 (t ).
(S5)
Under this boundary condition, Uk(t) can be determined from the values of U(t|z) at
k = 0, …, n-1, and the other way, the values of U(t|z) can be interpolated from the values
of Uk(t) at k = 0, …, n-1.
Uk(t) is calculated as a Euler-like equation of the time
6
evolution 7;.
U k (t  t )  U k (t )  t
k BT
F
  R  zmin
Gk 

z


,


(S6)
k  0,..., n  1
G has a form of a bi-weight function.
 48  z 2 2

1  ,
G ( z )   41 
4

0
|z|  2,
(S7)
otherwise.
It should be noted that the integral of G is not 1 but (48/41)×(16/15) = 768/615 ≈ 1.249.
The coefficient of 48/41 is set so that the maximum of the kernel of
K  z    R   in
Eq. S1 is 1 as follows:
48  9
1
2
9
1
 G(k  i) B (i)  41  16  6  1 3  16  6   1,
(S8)
k
k
where i is an integer. Consequently, the energy added to the biasing potential with  ps
during the time of t ps is 1.249kBT×(t/) kcal/mol. The force coming from U(t|) was
calculated as;
n
U (t | z )

 z  zmin 
  U k (t ) Bk 

z
z  z 
k 1
n 1
 z  zmin 
  U k' (t ) Bk' 
,
 z 
k 1
(S9)
where Bk' is the second order B-spline function:
( z  i  1) 2 / 2,

( z  i )( z  i  1)  1/ 2,
Bi' ( z )  
2
( z  i  2) / 2,
0,

-1  z  i  0,
0  z  i  1,
(S10)
1  z  i  2,
otherwise.
It should be noted that Bk' ( z ) is not the derivative of Bi(z).
7
U k' (t ) which corresponds to
the coefficient of the second-order B-spline expansion is given as:
U k' (t ) 
U k 1 (t )  U k (t )
.
z
(S11)
The ABMD method can be combined with the replica exchange method 8.
The
temperatures Tα and T, and the biasing potentials of Uα(t|z) and U(t|z) at the replicas α
and  are exchanged according to an exchange probability given by 7:
0
1
w( ,  )  
exp( )
0
(S12)
 1
1  
1


E  E 
U  z   U  z

 
k BT 
k BT 
 k BT


  
    k 1T U  z   U  z   ,





B
where Eα and E are ordinary atomic potential energies at the replicas α and ,
respectively. Using the ABMD simulations with the replica exchange method, kBT in
Eq. S6 was set at one to normalize the change of the biasing potential for all the replica
temperatures.
In this study, 96 equilibrated replicas for each system with an exponential
temperature distribution in the range 318 K ≤ T < 470 K were used.
The temperature
of 318 K was chosen to match the experimental conditions employed in the
measurement of the rate of protein degradation 9.
The distribution of the temperatures
was obtained using a web-server (http://folding.bmc.uu.se/remd/) so that the exchange
rate is ~ 0.135 at the desired temperature interval 10:
8
318.00, 319.37, 320.74, 322.12, 323.50, 324.89, 326.28, 327.68, 329.08, 330.48, 331.89,
333.30, 334.72, 336.15, 337.57, 339.01, 340.44, 341.88, 343.33, 344.78, 346.24, 347.70,
349.16, 350.61, 352.09, 353.56, 355.05, 356.53, 358.03, 359.53, 361.03, 362.54, 364.05,
365.56, 367.08, 368.61, 370.12, 371.66, 373.20, 374.74, 376.29, 377.85, 379.41, 380.98,
382.55, 384.12, 385.70, 387.29, 388.89, 390.48, 392.08, 393.69, 395.30, 396.91, 398.53,
400.16, 401.79, 403.43, 405.07, 406.72, 408.38, 410.04, 411.70, 413.37, 415.05, 416.73,
418.41, 420.11, 421.80, 423.51, 425.21, 426.93, 428.65, 430.37, 432.11, 433.84, 435.58,
437.33, 439.08, 440.84, 442.61, 444.38, 446.15, 447.93, 449.72, 451.52, 453.32, 455.13,
456.94, 458.75, 460.58, 462.40, 464.24, 466.08, 467.93, 469.78
Langevin dynamics algorithm was utilized to control the temperature and pressure of
the system. The coupling times for the temperature and pressure control were both set
at 2 ps-1.
The SHAKE algorithm
involving hydrogen atoms.
11; 12
was used to constrain all the bond lengths
The leap-frog algorithm with a time step of 2 fs was used
throughout the simulation to integrate the equations of motion.
Met1–Leu21 of the
seven N-terminal tails, Tyr123–Arg130 of the α-annulus and His99–Asn104
corresponding to a part of the seven activation loops in the activator were set to be free.
The other heavy atoms in the models were subjected to a 10-kcal/mol/Å2 harmonic
restraint in their original positions to prevent the structure from collapsing at a high
9
temperature.
To prepare for the 96 equilibrated replicas, 48 odd-number systems in which the
substrate was located in the activator and 48 even-number systems in which the
substrate was located in the antechamber of the CP were used.
First, the replica
system was heated from 0 K to 318 K within 1 ns during which the flexible parts in the
models and ions were fixed with decreasing restraints and the water molecules were
allowed to move.
After these the restraints were removed, the system was equilibrated
for 10 ns at a constant pressure of one bar and a temperature of 318 K with no restraint
(except to maintain the substrate within the range of between z = -23 and 37 Å and to
maintain the atoms in the models which should be fixed).
for the following REMD simulations.
Then the box size was fixed
Using the box size at 318K, the other 95 replica
systems were independently heated from 0K to their respective replica temperatures and
equilibrated for 10 ns at a constant volume at their respective replica temperatures.
The ABMD simulations were carried out at a constant volume with replica-exchanges
attempted every 125 steps (every 250 fs) for 300 ns per replica (a total of 28.8 μs for
each model).
The range of the reaction coordinate for the ABMD simulations was set
at -28 Å ≤ z ≤ 42 Å.
To sample data efficiently within the range, two harmonic
potentials with a force constant of 10.0 kcal/mol were applied at z = -28 Å and 42 Å.
10
As the free-energies around z = -28 Å and 42 Å may not be very reliable possibly due to
the introduction of the harmonic potentials, the range of z analyzed was limited to
within -23 Å ≤ z ≤ 37 Å. The resolution of the reaction coordinate, z, was set at
0.25 Å. The relaxation time for the free-energy profile in Eq. S1, , was set at 5 ps,
10 ps and 20 ps for 100 ns, respectively.
Consequently, the energies added to the
biasing potential per the range of 70 Å (from -28 to 42 Å) for 1 ns using  = 5, 10 and
20 ps were 2.29, 4.58 and 9.15 kcal/mol, respectively. The free-energy landscape, the
biasing potential times -1, fluctuates during the ABMD simulation.
Especially in this
study, sporadic local deformations of the free-energy landscapes were frequently
observed at almost any place during the simulations and the free-energy landscape
changes continuously because the substrate was often trapped locally inside the
complex (data not shown). Therefore, the free-energy landscape was estimated to be
an averaged free-energy landscape obtained from the last 100 ns simulation with
 = 20 ps.
The conformation of the complex for the analysis were stored every 1 ps
from the last 200-ns of the simulation.
The free-energy component analysis
The free energy of ΔG(z,T) was obtained from the ABMD simulation.
11
However, in
this study only the shape of ΔG(z,T) from z = -23 to 37 Å was estimated although the
value of ΔG(z,T) is generally set at zero at z = ±∞. Therefore, the difference between
ΔG(z,T1) and ΔG(z,T2) at different temperatures is unknown from the limited
information.
To obtain the difference, it has been assumed that the system in this
study follows the thermodynamic formula:
G ( z , T )  H ( z , T )  T S ( z , T )
T
H ( z, T )  H 0 ( z, T )   CV ( z, T )dT
(S13)
T0
CV ( z , T )
S ( z, T )  S0 ( z, T )  
dT
T0
T
 d CV 
CV ( z , T )  CV0 ( z , T )  (T  T0 ) 

 dT  0
T
where ΔH, ΔS and Cv are enthalpy, entropy and the heat capacity at a constant volume,
respectively. Then ΔG, ΔH and ΔS can be written as polynomial equations in terms of
T such as:
Gfit ( z, T )  1 ( z )   2 ( z )T   3 ( z )T log T   4 ( z )T 2
H fit ( z, T )  1 ( z )   3 ( z )T log T   4 ( z )T 2
(S14)
T Sfit ( z , T )   2 ( z )   3 ( z )  T   3 ( z )T log T  2 4 ( z )T 2
where αj (j = 1, …, 4) are parameters to fit the curves of ΔGfit, ΔHfit and ΔSfit to ΔG, ΔH
and ΔS, respectively.
It should be noted that when ΔGfit(z,T) is obtained, ΔHfit(z,T)
and -TΔSfit (z,T) are simultaneously obtained from Eq. S14. Here a quantity of δg(z,T)
is introduced to ΔG(z,T) as follows,
G( z, T )  Gfit ( z, T )   g ( z, T )
(S15)
As the system was assummed to follow the thermodynamic formula in Eq. S13, δg(z,T)
12
is regarded as an error which deviates from the polynomial equations of
Gfit ( z, T ) .
To
minimize δg(z,T), Eq. S15 was iteratively calculated as follows:
Eq. S15 is rewritten as G i ( z, T )  Gfiti ( z, T )   g i ( z, T ) in the i-th iteration.
(A) ΔG0(z,T) at each temperature was set so that the average of Δ0G(z,T) in the range of
-23 and 37 Å was zero.
Then, the parameters, αj (j = 1, …, 4) were calculated to fit
the curve in Eq. S14 to ΔG0(z,T), in which δg0(z,T) was fixed at zero.
(B)  g i ( z, T ) and its average in the range of z between -23 Å and 37 Å,  g i (T ) , were
calculated as follows:
 g i ( z, T )  G i ( z, T )  G ifit ( z, T ),
 g i (T )  
z 37
z 23
 g i ( z, T )dz / 
z 37
z 23
(S16)
dz.
(C) ΔG i+1(z,T) was reset to be Gi ( z, T )   g i (T ) so that the deviation between
ΔG i+1(z,T) and Gfiti ( z , T ) along z,

z 37
z 23
 G
i
fit

( z, T )  Gi 1 ( z, T ) dz , is minimized.
(D) The iteration was repeated from (A) and the parameters, αj (j = 1, …, 4) were
re-estimated to obtain the new fitting curves in Eq. S14 to the renewal ΔGi+1(z,T)
This cycle was repeated four times so that
Gfiti 1 ( z , T )  Gfiti ( z , T )  5.0 104 kcal/mol
(S17)
H fiti 1 ( z , T )  H fiti ( z , T )  3.5 103 kcal/mol
T Sfiti 1 ( z , T )  T Sfiti ( z , T )  3.5 103 kcal/mol
were fulfilled at any z and T. Thus, the unknown quantities of ΔG(z,T1) - ΔG(z,T2) can
13
be estimated to be ΔGfit(z,T1) - ΔGfit (z,T2). Finally, the curves of ΔH(z,T) and
-TΔS(z,T) were smoothed using a Bézier curve.
Reference simulation for a free substrate
An additional simulation of the substrate in bulk water was performed as a reference
for comparison with the distributions of the peptide in free space and confined space in
the complex.
The distribution of the temperatures was selected as:
318.00, 320.68, 323.38, 326.10, 328.84, 331.59, 334.36, 337.15, 339.95, 342.77, 345.62,
348.48, 351.36, 354.26, 357.17, 360.11, 363.06, 366.04, 369.03, 372.05, 375.09, 378.15,
381.22, 384.31, 387.42, 390.55, 393.70, 396.87, 400.06, 403.27, 406.50, 409.76, 413.03,
416.34, 419.65, 423.00, 426.36, 429.74, 433.16, 436.59, 440.04, 443.52, 447.02, 450.53,
454.08, 457.64, 461.23, 468.48
In this box, all the atoms of the substrate were separated more than 20 Å from the
edge of the box of 53 Å × 55 Å × 60 Å.
To neutralize the system, 2 chloride ions were
added. 13 sodium ions and 13 chloride ions were added to the box at a concentration
of 100 mM NaCl.
The protocol for this REMD simulation was the same as that for the
substrate in the complex. The REMD was carried out for 120 ns and the last 100-ns of
the trajectory was analyzed. The result showed that the radius of gyration of the
14
substrate at 318 K was 7.62 ± 1.22 Å in bulk. The average numbers of residues
forming -helix, 310-helix, -strand, turn and random coil in the 9-residue substrate
were 1.38 × 10-2, 1.79 × 10-1, 8.00 × 10-5, 2.33 and 5.58, respectively.
This indicates
that the substrate was naturally unfolded.
FIGURE LEGENDS
Fig. S1. A snapshot of the front N-gate closed-state observed in the PA26 – CPO
complex is shown.
Two cavities, represented with dots in blue and pink, were
analyzed by GHECOM.
The grid size of the dots is 2.0 Å. The blue and pink dots
which are closest together are shown as bigger dots.
The depth of the N-gate, which is
defined as the distance between these two bigger dots, is 2 5 Å in this figure. The
position of the N-gate, which is defined as the middle point between the two bigger dots,
is depicted as a cross.
The circle at the position of the N-gate with a radius of half the
depth of the N-gate and the circle at the center of mass of the substrate (z = 27.7 Å) with
a radius of gyration (Rg = 6.35 Å) are shown in thin and thick circles, respectively.
Residues 1 – 9 of the N-termini tails are depicted as traces of their Cα atoms. Tyr8 of
the N-termini tails and Gly128 of the α-annulus and the substrate are depicted as wire
models in red, blue and black, respectively.
15
Fig. S2. The root mean square deviations (RMSDs) of the free-energy landscape
between the free-energy landscape observed during the simulation and an averaged
free-energy landscape obtained from the last 100 ns with  = 20 ps are plotted against
the simulation time.
Fig. S3. The volumes of motion for the substrate are plotted against z.
Fig. S4.
The average values and RMSDs of the radius of gyration of the substrate are
plotted against z.
Fig. S5. The logarithm of the distributions of Tyr8 (Gly8) in the ordered- (disordered)
gates with regards to (r, z) is plotted.
The minimum value is set at zero.
distribution of residue Tyr126 at the α-annulus is shown as a reference.
The
For (d), the
data of Gly8 at (7 Å ≤ r ≤ 8 Å, -3 Å ≤ z ≤ 1 Å) and (8 Å ≤ r ≤ 9 Å, -2 Å ≤ z ≤ 1 Å) were
omitted for the clarity of the graph.
Fig. S6. 2-dimentional map of the number of contact pairs for the atoms of the interior
16
of the complex with the atoms of the substrate is depicted. The positions of all the
heavy atoms of the residues, whose total number of contact pairs with the substrate
contributed to more than 10% of the number of pairs of contacting atoms at any z in Fig.
6, are plotted. The residues at z ≤ 14 Å are from the CP, while those at z > 14 Å are
from the activator.
Tyr123–Arg130 of the CP form a part of the α-annulus, and
His99–Asn104 (Glu102 in PA26 and Ala102 in the PAN-like) of the activator form a
part of the activation loop.
The residues encircled by a dotted line are the hydrophobic
patch on the inner wall of the antechamber, while the residues encircled by a broken line
are those which were observed only in the disordered-gate models.
the graph, the all values of more than 0.2 were set at 0.2.
For the clarity of
In contrast to the
ordered-gate models, there was contact with the substrate mainly with Tyr26 of the
α-ring at z = ~ 7 – 17 Å in the disordered-gate models.
This is because Tyr26, which
interacted with Tyr8 in the ordered-gate models, could not interact with Gly8 in the
disordered-gate models and was exposed to the access of the substrate.
Fig. S7. The average values and RMSDs of the position of the N-gate are plotted
against z.
REFERENCES
17
1.
Ishida, H. (2011). Molecular Dynamics Simulation System for Structural Analysis of
Biomolecules by High Performance Computing. Prog. Nucl. Sci. Technol. 2, 470-476.
2.
Ishida, H. & Hayward, S. (2008). Path of Nascent Polypeptide in Exit Tunnel
Revealed by Molecular Dynamics Simulation of Ribosome. Biophysical J. 95,
5962-5973.
3.
Ishida, H. (2010). Branch Migration of Holliday Junction in RuvA Tetramer
Complex Studied by Umbrella Sampling Simulation Using a Path-Search Algorithm.
J. Comput. Chem. 31, 2317-2329.
4.
Hornak, V., Abel, R., Okur, A., Strockbine, B., Roitberg, A. & Simmerling, C. (2006).
Comparison of multiple amber force fields and development of improved protein
backbone parameters. Proteins 65, 712-725.
5.
Jorgensen, W. L., Chandrasekhar, J., Madura, J. D., Impey, R. W. & Klein, M. L.
(1983). Comparison of simple potential functions for simulating liquid water. J.
Chem. Phys. 79, 926-935.
6.
Deserno, M. & Holm, C. (1998). How to mesh up Ewald sums. I. A theoretical and
numerical comparison of various particle mesh routines. J. Chem. Phys. 1998,
7678-7693.
7.
Babin, V., Roland, C. & Sagui, C. (2008). Adaptively biased molecular dynamics of
free energy calculations. J. Chem. Phys. 128, 134101.
8.
Sugita, Y. & Okamoto, Y. (1999). Replica-exchange molecular dynamics method for
protein folding. Chem. Phys. Lett. 314, 141-151.
9.
Religa, T. L., Sprangers, R. & Kay, L. E. (2010). Dynamic Regulation of Archaeal
Proteasome Gate Opening As Studied by TROSY NMR. Science 328, 98-102.
10.
Patriksson, A. & van der Spoel, D. (2008). A temperature predictor for parallel
tempering simulations. Phys. Chem. Chem. Phys. 10, 2073-2077.
11.
Ryckaert, J., Cicotti, G. & Berendsen, H. J. C. (1977). Numerical integraton of the
Cartesian equations of motion of a system with constraints: molecular dynamics of
n-alkanes. J. Comput. Phys. 23, 327-341.
12.
van Gunsteren, W. F. & Berendsen, H. J. C. (1977). Algorithms for macromolecular
dynamics and constraint dynamics. Mol. Phys. 34, 1311-1327.
18