Polymer collapse in the presence of hydrodynamic interactions

5 Pages·0.12 MB·English

by  N. Kikuchi

Polymer collapse in the presence of hydrodynamic interactions
N. Kikuchi, A. Gent and J. M. Yeomans
Department of Physics, Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, England.
(February 1, 2008)
We investigate numerically the dynamical behaviour of a polymer chain collapsing in a dilute
solution. The rate of collapse is measured with and without the presence of hydrodynamic inter-
2 actions. We find that hydrodynamic interactions both accelerate polymer collapse and alter the
0 folding pathway.
0 PACS: 83.80. Rs
2

n I. INTRODUCTION and Yethiraj [16] compared molecular dynamics simu-
a lations of a polymer in a solvent to Brownian dynam-
J
ics simulations. They attempted to match the param-
5 When a polymer is placed in solution hydrophobic in- eters in the two simulations and hence to compare col-
1
teractions between monomers and solvent molecules can lapse with and without hydrodynamics. Here we use a

] cause it to undergo a collapse transition to a compact hybrid approach [17] where the solvent is modelled by
ft state [1,2]. The statistical physics of the polymer tran- a Malevanets-Kapral method [18] and the polymer by
o sition from the extended to the collapsed state is well molecular dynamics in our investigation of the hydrody-
.s understood. Howeverthe dynamicsofthe transitionand namics of polymer collapse.
t inparticularthe effectonthatdynamicsofthe hydrody-
a
m namic properties of the solvent remains unclear. There-
foreinthisLetterweinvestigatenumericallythedynam- II. SIMULATION DETAILS
-
d ical behaviour of a polymer chain collapsing in a dilute
n solution. The collapse is measuredwith and without the Modelling a dilute polymer solution is a difficult task
o
c presence of hydrodynamic interactions thus allowing a because of the existence of widely differing time scales.
[ direct investigation of their effect. We find that hydro- The dynamicalpropertiesofpolymers canbe dominated
dynamics accelerates the polymer collapse. It also alters by hydrodynamic interactions between different parts of
1
v the folding pathway, allowing the folding to occur more the polymer chains [2]. In contrast with the time scale
6 homogeneously along the polymer chain rather than ini- ofthermalfluctuationsofindividualmonomers,thesein-
4 tially at the chain ends. teractions are long-ranged and evolve slowly. Therefore
2 With the introduction of Zimm’s model [2] it became it is computationally too expensive to reach hydrody-
1
apparent that hydrodynamics play a central role in the namic time scales using molecular dynamics simulations
0
2 dynamics of polymers in dilute solution. However, un- for both the polymer and the solvent molecules.
0 derstandingsuchinteractionsisdifficult, analyticallybe- To overcome this problem we use a hybrid simulation
/ cause they present a complicated many-body problem, approach where the equations of motion of the polymer
t
a andnumericallybecausetheydevelopontime-scaleslong alone are solved using a molecular dynamics algorithm.
m
compared to the thermal fluctuations of the monomers. The solvent is modelled using a mesoscale approach,de-
- Theoreticalwork on the dynamics of polymer collapse veloped by Malevanets and Kapral [18]. This ignores
d
can be divided into two approaches. Phenomenological the molecular detail of the solventbut preservesits abil-
n
o models balance the driving and dissipative forces to give ity to transmit hydrodynamic forces. The polymer can
c scaling laws [3–7]. They involve assumptions about how be thought of as moving within a “hydrodynamic heat
:
v the collapsed state develops on which there is no con- bath”.
i sensus. Several authors have considered models which Thepolymerchainismodelledbybeadsconnectedvia
X
are based on a solution of the Langevin equation [8–10]. non-harmonicsprings[19]with adjacentbeads along the
r
a Of particular interest is work by Pitard [11] and by chain backbone representing an effective Kuhn length of
Kuznetsov et al [12] who find the inclusion of hydrody- the polymer chain. These finitely extensible springs are
namics, modelled by a preaveragedOseen tensor, speeds represented by the FENE potential
up the collapse.
2
Simulations on polymer collapse [4,13–15], using VFENE(r)=−κR02ln 1− r , r<Ro. (1)
Monte Carlo or Langevin approaches, have not included 2 " (cid:18)R0(cid:19) #
thehydrodynamiceffectsofthesolvent. Veryrecentwork
has shown that it is now possible to use molecular dy- A Lennard-Jones potential [20] which acts between all
namics simulations with anexplicit solventto model the the polymer beadsis usedto modelthe excludedvolume
collapse transition if powerful computational resources ofthemonomersandalongrangeattractionwhichdrives
are available [7]. In an interesting recent paper Chang polymer collapse
1
σ 12 σ 6 relaxed rapidly (t ≤ 100) to the equilibrium Maxwell-
V (r)=4ε − . (2) s
LJ
r r Boltzmann form.
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
Aparticularlyusefulfeature ofthe Malevanets-Kapral
We take ε = 1.0, σ = 1.0, κ = 30 and R0 = 2 where
algorithm is the ease with which hydrodynamic interac-
parameters and results are quoted in reduced Lennard-
tions can be “turned-off” thus replacing the hydrody-
Jones units.
namic heat bath by a Brownian (random) heat bath.
Newton’sequationsofmotionforthepolymerareinte-
This is achieved by randomly interchanging the veloci-
gratedusingthetimereversiblevelocityVerletalgorithm
ties of all the solvent particles after each collision step,
[20]. The molecular dynamics time step is chosen to be
thus relaxing the constraint of local momentum conser-
δt=0.002t wheret is the intervalbetweensolventcol-
s s
vation to a global one. Accordingly the velocity corre-
lision steps, defined below.
lations which result in hydrodynamic interactions disap-
ThesolventismodelledbyalargenumberN =131072
pear from the fluid. Running simulations with the same
of point-like particles which move in continuous space
initial conditions and parameter values, but with hydro-
withcontinuousvelocitiesbutdiscretelyintime[18]. The
dynamicspresentorabsent,greatlyfaciliatespinpointing
algorithm is separated into two stages. In the first of
the effect of the hydrodynamic interactions.
these, a free streaming step, the positions of the solvent
particlesattime t, x (t), areupdated simultaneouslyac-
i
cording to
III. RESULTS
x (t+t ) = x (t)+v (t)t (3)
i s i i s
where v (t) is the velocity of a particle. Our aim is to study the dynamics of polymer collapse
i
from an extended to a compact state [1]. The collapse
The second component of the algorithm is a collision
trasnsition is driven by the attractive Lennard-Jones in-
stepwhichisexecutedonbothsolventparticlesandpoly-
mer beads. The system is coarse-grained into L3 unit teractions between the polymer beads. The transition is
roundedandshiftedbythe finite chainlength, andtakes
cells of a regular cubic lattice. In this simulation L=32
place at k T ∼1.8 for chains of length N =100.
isused. Thereisnorestrictiononthetotalnumberofsol- B p
Initial polymer configurations were chosen at random
vent or polymer particles in each cell, although the total
from long runs on a polymer chain at equilibrium in a
number of particles is conserved. Multiparticle collisions
solvent with k T = 4. Each of these extended con-
are performed within each individual cell of the coarse- B
figurations were placed in a solvent at equilibrium at
grained system by rotating the velocity of each particle
relative to the centre of mass velocity v (t) of all the kBT =0.8. This value ensures collapse but prevents the
cm
chain collapsing so quickly that hydrodynamic interac-
particles within that cell
tions do not have sufficient time to develop. The rate of
vi(t+ts) = vcm(t)+R(vi(t)−vcm(t)). (4) polymer collapse was measuredby monitoring the varia-
tion of the radius of gyration R of the chain with time
R is a rotation matrix which rotates velocities by θ g
around an axis generated randomly for each cell and at
eachtimestep. Inthepresentcalculationswetakeθ = π2. Rg2(t)= 1 Np (Ri(t)−Rcm(t))2,
Note that the collision step preserves the position of Np i=1
X
the solvent and polymer beads. It transfers momentum
1 Np
between the particles within a given cell while conserv-
R (t)= R (t). (5)
cm i
ingthetotalmomentumoftheseparticles. Becauseboth Np i=1
momentumandenergyareconservedlocallythethermo- X
hydrodynamic equations of motion are captured in the AtypicalnumericalresultisshowninFigure1forachain
continuum limit [18]. Hence hydrodynamic interactions of length Np =100 with and without hydrodynamics.
can be propagatedby the solventand, because the poly- The collapse time was estimated as the time when the
mer beads are involved in the collisions, to the polymer. equilibriumradiusofgyrationwasfirstattained. Results
Note, however, that molecular details of the solvent are for τ are shown for chains of varying lengths in Table 1.
excluded: this allows the hydrodynamic interactions to τ wasaveragedover20initialconfigurationsforN =40,
p
be modelled with minimal computational expense. 60 and 100, and 5 for N = 200, and the collapse time
p
The volumeinphasespaceis invariantunder boththe for hydrodynamic (τ ) and Brownian (τ ) heat baths
H B
free streaming and collision steps. Hence the system is compared. Variations in collapse times between differ-
describedbyamicrocanonicaldistributionatequilibrium ent runs are large as expected. However, it is strikingly
[18]. The initial solvent distribution was generated by apparent that hydrodynamic interactions speed up the
assigning positions randomly within the system with an rate of collapse by a factor∼2 for each polymer length.
average density ρ = 4 particles per unit cell. The ve- For all but 6 of the 65 runs and for all the N = 100
p
locities were assigned from a uniform distribution which and200runsthecollapsewasfasterwithhydrodynamics
2
switched on. The results in Table 1 also illustrate that chain may be important. Moreover collapse is driven by
thecollapsetime increaseswithchainlengthasexpected attractive Lennard-Jones interactions between the poly-
(τ ∼N1.6±0.2, τ ∼N1.2±0.3). Values predicted in the mer beads rather than repulsive monomer-solvent inter-
B p H p
literature for the exponents relating the collapse time to actions. It would be of interest to include the solvent
chain length vary widely. It is far from obvious that nearthe polymer inthe morerealistic moleculardynam-
these exponentsareuniversal: they appeartodepend on ics updating and the possibility of doing this is under
quench depth and the details of the model [16]. More- investigation.
over the final radius of gyration was recorded for both Finally it may be of interest to consider whether the
the Brownian and hydrodynamic collapses and found to earlystagesofthefoldingofproteinsinaqueoussolution
agree as the equilibrium polymer properties should be mightbe influencedbythe hydrodynamicinteractionsof
independent of the nature of the heat bath. the solvent increasing the cooperativity of the collapse.
Figure 2 compares typical collapse pathways with and Acknowledgements We thank A. Malevanets, C.
withouthydrodynamicsforN =100andafinaltemper- Pooley and P. Warren for helpful discussions.
p
aturek T =0.8. Inthisandthemajorityofotherrunsa
B
qualitative difference was observedin the collapse mech-
anism. In the Brownian solvent the ends of the polymer
tendtocollapsefirstformingadumbbellshape. Withhy-
drodynamics the collapse takes place more evenly along
the chain. We speculate that this occurs due to ’slip-
[1] P. G. de Gennes, Scaling Concepts in Polymer Physics,
streaming’. The movement of beads towards each other,
Cornell University Press, New York,(1979).
whichisinitiatedbytheattractiveLennard-Jonespoten-
[2] M. Doi and S. F. Edwards, The theory of polymer dy-
tial, is enhanced by hydrodynamic interactions. Once a
namics, Clarendon Press, Oxford, (1986).
beadstartsmovinginagivendirectionandlocallydrags [3] P. G. de Gennes, J. Physique. Lett. 46: L-639 (1985).
fluid with it, it is easier for neighbouring beads to move [4] B. Ostrovsky and Y. Bar-Yam, Europhys. Lett. 25: 409
in the same direction. Similar behaviour albeit on much (1994).
largerlengthscaleshas been observedexperimentally by [5] A. Buguin, F. Brochard Wyart and P. G. de Gennes,
Bartlett et al [21] for a pair of colloidal particles in solu- Comptes Rendus de L’Academie des Sciences IIB 322:
tion. 741 (1996).
[6] A. Halperin and P. M. Goldbart, Phys. Rev. E 61: 565
(2000).
IV. DISCUSSION [7] C. F. Abrams, N. Lee and S. Obukhov, cond-
mat/0110491 (2001).
[8] E. G. Timoshenko and K. A. Dawson, Phys. Rev. E 51:
We have shown that it is possible to measure the dy- 492 (1995).
namicsofthecollapsetransitionofapolymerinasolvent [9] E.PitardandH.Orland,Europhys.Lett.41: 467(1998).
usingahybridmesoscale/moleculardynamicsalgorithm. [10] F. Ganazzoli, R. La Ferla and G. Allegra, Macro-
Hence it has been possible to show that hydrodynamic molecules. 28: 5285 (1995).
interactions speed up the collapse of the polymer chain. [11] E. Pitard, Eur. Phys. J. B. 7: 665 (1999).
[12] Y. A. Kuznetsov,E. G. Timoshenko and K. A. Dawson,
The hydrodynamics alters the collapse pathway, allow-
J. Chem. Phys. 104: 3338 (1996).
ing the folding to occur more homogeneously along the
[13] A. Byrne, P. Kiernan, D. Green and K. A. Dawson, J.
chain, rather than initially at the chain ends.
Chem. Phys.102: 573 (1995).
Qualitative features of the collapse pathways are in
[14] Y. A. Kuznetsov,E. G. Timoshenko and K. A. Dawson,
agreement with those reported by Chang and Yethiraj J. Chem. Phys. 103: 4807 (1995).
[16] who recently compared molecular dynamics simu- [15] B. Schnurr, F. C. Mackintosh and D. R. M. Williams,
lations of a polymer in a solvent, which include hydro- Europhys. Lett. 51: 279 (2000).
dynamics, to Brownian dynamics simulations, which did [16] R. Chang and A. Yethiraj, J. Chem. Phys. 114: 7688
not. These authors also found that the Brownian simu- (2001).
lationscouldbecometrappedinametastablefreeenergy [17] A. Malevanets and J. M. Yeomans, Europhys. Lett. 52:
minimum. Itwouldbeinterestingtoaddresswhetherthis 231 (2000).
[18] A.MalevanetsandR.Kapral,J.Chem.Phys.110: 8605
is due to deeper quenches than those considered here or
(1999).
to the different simulation approaches employed.
[19] K. Kremer and G. S. Grest, J. Chem. Phys. 92: 5057
The solvent is modelled using the Malevanets-Kapral
(1990).
algorithm which sustains hydrodynamic modes but does
[20] M. P. Allen and D.J. Tildesley, Computer simulation of
not include molecular interactions. Hence we caution liquids, Clarendon Press, Oxford, (1989).
that we cannot investigate the late stages of collapse [21] P. Bartlett, S. I. Henderson and S. J. Mitchell, cond-
where trapping of the water molecules in the polymer mat/0012271 (2000).
3
FIG. 1. Variation of the radius of gyration with time for a collapsing polymer chain in a hydrodynamic (—) or Brownian
(···) heat bath.
FIG.2. A comparison of thepathwaysfor polymer collapse with and without hydrodynamics.
4
Np hτHi hτBi hRgHi hRgBi
40 290 ± 92 401 ± 118 2.02 ± 0.05 2.02 ± 0.04
60 490 ± 79 671 ± 170 2.23 ± 0.02 2.23 ± 0.01
100 858 ± 113 1571 ± 252 2.58 ± 0.01 2.58 ± 0.02
200 2050 ± 253 5500 ± 390 3.15 ± 0.00 3.16 ± 0.02
TABLEI. AveragedcollapsetimeinunitsofthesolventtimesteptsofapolymerchainoflengthNpwith(hτHi)andwithout
(hτBi)hydrodynamicsforafinaltemperaturekBT =0.8. TheradiusofgyrationaftercollapseinreducedLennard-Jonesunits
with (hRgHi) and without (hRgBi) hydrodynamicsis also listed.
5