Article pubs.acs.org/Macromolecules
Maximum Compaction Density of Folded Semiflexible Polymers Anna Lappala and Eugene M. Terentjev* Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, U.K. ABSTRACT: We study the dynamics of polymer chain collapse into a globular state in poor solvent, as a function of chain flexibility. We examine the compactness of the folded globule assessing the direct contact and a larger length-scale structural characteristics at various persistence lengths lp. We discover that semiflexible polymer chains with a specific stiffness (lp ≈ 8 monomers) form the most densely folded structures, independently of the chain length, a phenomenon due to nematic-like hairpin formation and stacking of hairpin segments in the most compact state. Even in this most compactly folded state, the number of contacts between monomers (accounting for covalent bonds as well as noncovalent physical interactions) is still low, and the globule is only just above the marginal stability threshold. We identify morphological changes associated with the dynamics of semiflexible chain collapse: flexible chains fold into globules, less flexible form rod-like structures, followed by toroidal structures of different geni, and finally, even stiffer chains form highly elongated rods. We also study the time of collapse as a function of persistence length, which shows that stiff chains take much longer to reach their elongated lowest energy state. We propose that polymer chain stiffness, as a regulator of both local and global chain compactness, is highly important in biological systems and in the dynamics of DNA and protein folding.
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INTRODUCTION The coil−globule transition is the collapse of a statistical coil state of a polymer chain (self-avoiding random walk) into a densely packed globular configuration in a response to a change from good to poor solvent conditions, e.g., as a result of temperature quenching.1 Theoretical and experimental studies together with computer simulations have shown that the process of coil−globule transition differs depending on the persistence length of a polymer chain; however, there are no systematic studies of the process of collapse dynamics as a function of polymer flexibility.2−4 Model Brownian-dynamics simulations presented here describe the behavior of a single homopolymer chain of varying flexibility. Long chain is studied (N = 2000) so that morphological features and the kinetics of the transition could be clearly identified. We quantitatively analyze the dependence of the collapse on chain flexibility by monitoring contact densities at different persistence lengths over time. Work described here follows our earlier work,4 where the stiffnessdependent morphological properties of polymer chains were recorded and classified in two groups: globular (equilibrium globule) state for flexible chains, and rod-like elongated hairpinbased forms for semiflexible chains. In this work, we show that even stiffer chains exhibit elongated rod and toroidal morphologies (Figure 1) that have been observed experimentally5 and in computer simulations.2 However, most published models used for simulations are rather complex, involving charged polymers,6 entanglement of several chains,7 spatial confinement of chains in channels/ spheres,8 physical confinement of polymers via potentials such © 2013 American Chemical Society
Figure 1. Configurations of individual polymer chains at different persistence lengths during the process of collapse. The range is between lp = 1 monomer (later in the text we shall refer to that lengthscale as σ) for a flexible chain and lp = 75σ for a stiff semiflexible chain.
as end grafting.9 Morphologically, the behavior of semiflexible chains has been studied by Monte Carlo simulations:10 this method allows one to overcome possible “trapping” in local minima of the free energy landscape, and even though it is impossible to track the dynamics of the process in this case, it is valuable to know what the global free energy minimum conformation of the polymer chain is, depending on its stiffness. In our work, the objective is to reduce the problem to Received: May 2, 2013 Revised: July 17, 2013 Published: August 21, 2013 7125
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the simplest possible scenario; in our simulations we focus on the dynamics of semiflexible chains over time, analyzing the data quantitatively. Earlier Brownian dynamics simulations of shorter chains have reported torus formation as well as rod (racquet) morphologies.2 However, the analysis of results, although elegant theoretically,2,3,11 does not incorporate all the different metastable states and oversimplifies the process of collapse itself. Brownian dynamics simulations that show toroidal configurations have been performed using numerous models, an example of which is a freely jointed square-well chain with a harmonic bending potential described by Noguchi et al.12 However, the analysis of Noguchi et al. is more centered on toroid formation, and the structural aspects of collapse as opposed to a more holistic representation of collapse that we present here. Collapse as we see it in several sets of our simulations is a process that is highly regulated by local as well as large-scale topological properties of chains, and hence one semiflexible chain may end up becoming a torus, whereas if the symmetry of a 2-hole torus state is broken, the chain might collapse into a rod, which does not exclude the possibility that a configuration of a similar genus might happily stay in its toroidal lowest energy state. In order to study the morphological properties of semiflexible chains, we analyze the number of intermonomer contacts within individual polymer chains of varying stiffness over time. We also study the dense (collapsed) lowest energy states, highlighting the morphology- and density profiles of collapsed chains for variable persistence lengths and investigate the evolution of collapse time as a function of persistence length. In addition, we relate our findings to biologically relevant processes. There are many examples of biologically important flexibility-dependent mechanisms, and in this work we discuss the compaction of the DNA in the nucleus and the relevance of loop formation in essential biological processes such as gene regulation and transcription. We also discuss the collapse of certain proteins and bio/polymers into toroidal morphologies.
Figure 2. The circles approximately represent the region of hard-core repulsion due to LJ potential, just over 0.5σ. The equilibrium length of a covalent bond between consecutive monomers on the chain is due to the combination of LJ and FENE potentials, rbond = 0.94σ for the chosen κ. The minimum of the LJ potential acting between noncovalently interacting monomers is at r0 ≈ 21/6σ, and we choose the radius of the shell representing monomers in direct contact as 1.1σ. The much bigger shell (2σ) captures many more monomers in order to process the average density. Angle θ represents the angle between monomers associated with the bending elasticity of the chain.
FENE potential is harmonic near its minimum, so that the effective spring constant between monomers is κ. However, a FENE bond cannot be stretched beyond a maximum length determined by R0 (see below for the discussion of numerical values choice). (2) The Lennard-Jones potential for nonconsecutive pairs of monomers: ULJ = 4ε[(σ/r)12 − (σ/r)6], where σ is the distance at which the interparticle potential is zero, thus separating the repulsive and attractive regions, and ε the potential well depth. This potential reaches its minimum ULJ = −ε at r = 21/6σ. For numerical simulations, it is common to use a shifted and truncated form of this potential, which is set to zero past a certain separation cutoff.4 For a self-avoiding random-walk chain (i.e., swollen in a good solvent), the potential is set to zero at rcutoff = 21/6σ, representing only the soft repulsion between monomers close-up: ⎧ ⎡ 1⎤ 12 6 1/6 ⎪ 4ε⎢(σ /r ) − (σ /r ) + ⎥ , r ≤ 2 σ ⎦ ⎣ 4 ULJ = ⎨ ⎪ 1/6 ⎩ 0, r > 2 σ
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SIMULATION MODEL The model used for numerical simulations in this work is based on the bead−spring model described in the moleculardynamics context by Kremer and Grest.13 The system was simulated in a standard way, with a fixed number of particles in a fixed volume of a cubic cell with periodic boundary conditions, which allows particles to exit and re-enter the simulation box. A polymer chain is composed of connected monomeric units consisting of N = 2000 monomers; we find this chain length sufficient to demonstrate the stiffnessdependent dynamics of individual polymer segments during the coil−globule transition. For simplicity, one can assume that each monomer has a diameter of σ; strictly speaking, σ is not exactly the “size” of a monomer, but it is close to the equilibrium chain spacing; monomers in contact are at distance rmin = 0.94σ, see Figure 2. Interactions between monomers are described by the following standard potentials: (1) Finitely extensible nonlinear elastic (FENE) potential for connected monomers (residues): UFENE
⎧ 1 2 ⎪− κR 0 ln[1 − (r /R 0)2 ], r ≤ R 0 =⎨ 2 ⎪ 0, r > R · ⎩ 0
(2)
In order to simulate poor solvent conditions when there is an effective attraction between monomers, the cutoff value was set to for rcutoff = 3σ. In each case, the potential is also shifted up or down to ensure the continuity at the cutoff radius. (3) Bending elasticity of a semiflexible chain is described by a potential Ustiff = Kθ (1 − cos θ), where θ the angle formed between two consecutive bonds, and Kθ is the corresponding bending constant. In Figure 2, we present a schematic diagram of intermonomer bonds as well as local contacts that are crucial for further density and contact map analysis. For simulated chains for N = 2000 monomers, parameters used for simulations are as described in ref 14 with the monomer size σ chosen to be 0.3 nm (which is a typical size of an amino acid residue in proteins). Parameters for κ and R0 were set so as to avoid any bond crossing. For our simulations, the values were used as in ref 13: the maximum bond length R0 = 1.5σ and the spring constant κ = 30ε/σ2. The spring constant was studied by Kremer and Grest13 and was found to be strong enough such that the maximum extension of the bond was always less than 1.2σ for kBT = 1.0ε, making the covalent bond breaking energetically unfeasible. The important ratio, Kθ/kBT,
(1) 7126
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Figure 3. (a) Contact density evolution over time for contacts within a radius of 1.1σ. (b) Number of contacts (within 1.1σ) in the collapsed state as a function of the chain persistence length. Error bars arise from repeating each simulation four times.
Figure 4. (a) Contact density evolution over time for contacts within a radius of 2σ. (b) Number of contacts (within 2σ) in the collapsed state as a function of persistence length. The similar trend with a clear maximum at lp = 8σ is seen, as in Figure 3b.
which is the measure of the chain persistence length (lp = σKθ/ kBT) was set to vary from lp = σ (or Kuhn’s length of 0.6 nm) up to lp = 75σ (a very stiff chain with the Kuhn’s length of 45 nm) depending on the flexibility of the simulated chain. Integration time step was chosen to be tint = 0.012τ, where τ is the reduced (Lennard-Jones) time, defined as τ = σ(m/ε)0.5, as discussed in refs 13 and 15. Taking the values of σ and ε above, and the typical mass of an aminoacid residue (molecular weight ≈100), we obtain the estimate of τ ≈ 0.03s. Hence, with 106 time-steps of this coarse-grained simulation, we are able to follow the dynamics of a polymer chain for 360 s. In order to study the coil−globule transition, the pairinteraction (Lennard-Jones) potential is altered: at t = 0 the “good-solvent” repulsive potential with rcutoff = 21/6σ is switched to the attractive potential with rcutoff = 3σ, which represents an instantaneous quenching of the system. We then simulate the process of collapse over time, as discussed in ref.4
For contacts within 1.1σ, we demonstrate that in the initial structure of the expanded swollen chain, one monomer is only contacting 2 neighboring monomers along the chain, as expected for a random coil. So the total of 3 monomers, multiplied by N = 2000, gives the starting value of 6000 in Figure 3a. After this, the process of collapse is followed over time: initially, the contact density increases rapidly as the structure collapses, and then after a certain period of time (which depends on the stiffness of the polymer) the chain finds its stable folded lowest free energy configuration, which is associated with a high contact density. The number of bonds in the lowest energy state is persistence-length dependent: the most dense state (lp = 8σ) has the highest density of neighbors: approximately 4 per monomer (this value is derived from the plateau of lp = 8σ shown in Figure 3, which is approximately at 10000 contacts; this number of contacts divided by the length of the chain, N = 2000 monomers, and excluding the self-contact, gives four neighbors per monomer). There is a clear difference between the plateau values for different persistence lengths; in Figure 3b, we look at the contact density in the final folded state for chains of increasing bending stiffness. In order to obtain these values, the simulations were performed for 24 million time-steps, which ensured that the plateau value is stable for over a million simulation steps. An interesting and perhaps unexpected feature of contact density as a function of persistence length is that there is a maximum at lp = 8σ, meaning that at this persistence length, the polymer chain is in its densest conformation. It is known that for random dense packing of spheres, the volume
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CONTACTS BETWEEN MONOMERS In order to understand the morphological properties of structures emerging during the coil−globule transition, we identified two contact types: (1) “bonds”, i.e., contacts within the radius of 1.1σ, and (2) “spatial affinity” within the radius of 2σ. The latter gives a measure of the average local density, while the bonding includes both the “covalent bonds” (represented by the FENE potential) and the “noncovalent contacts”, where the usual van der Waals or hydrophobic interactions are described by our LJ potential. 7127
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fraction is ϕc = 0.64. This value can be translated into the maximum possible number of contacts between spheres, known as the coordination number z, and for ϕc = 0.64, zc = 8.6 as discussed in refs 16 and 17. This value is much higher than the coordination number obtained from our simulations of polymer chains, z ≈ 4, meaning that polymer chains, even in their most compact folded state, are still very “open” structurally.
are listed; the radius of contact was chosen to be 2σ in this case, to represent the local packing properties of monomers. In the resulting contact maps, the main diagonal represents selfcontacts and off-diagonal contacts represent the morphology of local contacts between monomers that are not neighbors along the chain. More specifically, any two extended chain segments running parallel to each other within the contact radius are represented by a segment of straight line on a contact map, parallel to the main diagonal, or at 90° to the main diagonal when the segments are antiparallel. Any hairpin chain configuration is, therefore, represented by a line at 90° starting on the main diagonal. When two hairpins are folded close together, so that their extended segments are all in contact, we find a characteristic diamond shape at an off-diagonal position nm, between the hairpin at the nnth position and the one at the mmth position on the contact map. In Figure 6d we also see a number of half-diamonds, where only a part of each hairpin’s segments is in contact with the other. The length of straight lines on contact maps (whether the hairpins coming off the main diagonal, or the off-diagonal segments in extended contact) increases with the chain stiffness, which we measure by the persistence length lp = σKθ/kBT. The nature of this dependence in the dense collapsed state may not be a priori obvious, since interactions between monomers not linked by the chain are now highly important. For instance, the theory of hairpin folding of semiflexible chains with nematic interactions18 predicts a highly nonlinear relation between the length of extended segments and the bending stiffness (in our notations, one would expect Lseg ∝ lp exp(Kθ/ kBT)1/2.19 However, our statistical analysis of straight segment length on all the contact maps gives a conclusion that, in spite of the dense state of the chain and the increased role of crossmonomer interactions, the mean length of extended segments in parallel alignment remains linear with the original persistence length lp (Figure 7). Similarly to the contacts within the radius of 1.1σ, the persistence length of 8σ gives the most compact state, which is due to a phenomenon we refer to as hairpin stacking, in which hairpins of semiflexible chains are organized such that they are stacked on top of each other. The number of contacts monotonically decreases after reaching this maximum, and this is due to metastable toroidal structures of the polymer chain (which mostly are formed between lp = 30σ and lp = 40σ). In this case, locally, monomer packing density still remains rather high as a result of hairpin formation: using contact map analysis, we confirmed that hairpins are still present in the structure; however, globally, the contact density decreases because stiffer chains do not collapse into a single globule but rather stay in a torus configuration with a single hole as shown in Figure 6c, which obviously decreases the number of contacts on a global scale. We called the torus “metastable” here because in our longest simulations we are occasionally finding a transition from the torus to a rod of folded parallel hairpins, which then remain unchanged. We could not investigate the statistics of this transition, and hence address the intriguing question of the true ground state and the transition barrier between these two states (merely for limited computational resources), but it is nevertheless clear that since such transitions occasionally happen, and never a reverse transition from a rod to a torus, then the torus is not the final global equilibrium state.
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LOCAL PACKING OF MONOMERS For local density assessment, we examine the average number of monomers within a radius of 2σ. In this case, the initial unfolded chain configurations typically have five monomers within this sphere, i.e., four neighbors to each monomer along the chain. This gives the five monomers in total, times 2000, gives the starting value of 10000 in Figure 4a. This number increases up to 26.5 neighbors per monomer in the collapsed state, the estimate obtained using the same method of counting as in Figure 3a. This difference between local packing and covalent “bonds” shows the importance of local compaction properties of a chain: the high contact density in the folded state is a sign of high compaction; however, this value does not demonstrate how exactly the structures are packed. In our earlier work,4 we showed that contact maps can help one to understand and visualize the structural properties of a folded chain. It is interesting and important to examine the issue of the packing density maximum, seen in both representations: for the contact radius of 1.1σ in Figure 3b and of 2σ in Figure 4b. This maximum, broadly at lp ≈ 8σ, is clearly only possible for long chains: when the number of residues N becomes low it would be difficult to densely pack a semiflexible chain. We thus examine the universality of this maximum by studying chains with N = 1000 and 3000, in otherwise the same conditions, and comparing their packing density with the earlier results for N = 2000: Figure 5 illustrates that for long chains the maximum at the characteristic value of persistence length remains universal. Contact maps represent intermonomer three-dimensional connectivity profiles in a two-dimensional map (Figure 6). For each consecutive monomer, all contacts within a given radius
Figure 5. An illustration of the maximum density at lp ≈ 8σ for different chain lengths, N = 1000, 2000, and 3000. The contacts are counted within the radius 2σ as in Figure 4; their number is scaled by N in order for the three curves to appear in the same scaling range. The (middle) data points for N = 2000 are the same as presented in Figure 4(b). 7128
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Figure 6. Contact maps for a contact radius of 2σ of metastable configurations for lp = 40σ (a−c) and lp = 75σ (d).
It is interesting to note that the stretched exponential relaxation with an exponent of 2/3 is usually a signal of a simple Gaussian distribution of relaxation times. If one has a combination of many simple relaxation processes, each represented by the exponential with its own relaxation time Δ, the resulting average (observable) process would be given by the equation f (t ) = const ·
∫0
∞
2
e−t / Δe−Δ /2sdΔ
≈ const ·exp[−(t / s )2/3 ]
(3)
Figure 7. The average length (in number of monomers) of the straight segments on contact maps of collapsed chains, representing the extended chain segments in close parallel alignment, is plotted against the bending stiffness expressed via the thermal persistence length lp.
assuming the Gaussian probability P(Δ) ∝ exp[−Δ2/2s] and estimating the integral via the steepest descent. This gives the observed (fitted) relaxation time τ ≈ √s, that is, determined by the width of the Gaussian distribution of relaxation times. The fitted values of the time to collapse τ as a function of persistence length, are shown in Figure 8. From these results it is obvious that the time to collapse has the highest rate of increase with lp around the same point where we find the densest globule packing. In the interval between lp = 10−30σ,
DYNAMICS OF COLLAPSE The contact density of polymer chains is not the only valuable information that we obtain from our data of contact densities versus time (as shown in Figures 3 and 4) . We notice that the rate of the folding process is also dependent on the persistence length of the chain: more flexible chains tend to collapse more rapidly, whereas stiffer chains take longer to reach the state of equilibrium. In order to extract the time that it takes for a polymer to collapse, the data for contact density for each persistence length as a function of time, as shown in Figure 4, were fitted with a stretched function of form f(t)∝ e(−(t/τ)β), where β = 2/3. For appropriate fitting, initial and final number of contacts were set independently for each lp, and then values for τ and β were fitted; however, it turns out that β ≈ 0.67 is legitimate for all curves and thus only one parameter (the relaxation time τ) had to be fitted in the second round.
Figure 8. Time to collapse as a function of persistence length. The sigmoidal solid line is a guide to an eye.
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the rate of collapse increases much slower compared to flexible chains: it is in this region where we observe the transition from a tightly packed elongated structures to toroidal intermediate states. Above lp = 30σ, the collapse rate seems to stabilize, and morphologically, in this region we observe the transition from toroid-like intermediates to elongated thin rod morphologies. Even though the relaxation function f(t) appears to plateau at a certain value, this is obviously not true as for an infinitely stiff chain the time to collapse will also become infinity; therefore, we expect a slow gradual increase for values higher than those in the spectrum of our simulations.
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HAIRPIN LOOPS AND TOROIDS IN BIOLOGY In a single eukaryotic cell, two meters of DNA is packed into a highly structured and compacted state in the cell nucleus, which has a diameter of ∼10 μm. The genetic material in the cell can be compacted into two possible morphologically and functionally distinct states: (1) euchromatin, a flexible and active morphology of the genome in which genes are actively expressed to produce functional proteins, and 2) heterochromatin, a more compact and inactive state of the genetic material which is found in chromosome regions where genes are not expressed, i.e., “silent”. This compaction-dependent functionality is very important in determining whether certain genes or gene groups are functional at any given time or if these genes are not expressed (heterochromatic). This is often seen during the development of an organism: over time, certain genes will be turned “on” or “off” in a highly regulated fashion.20 Another interesting example is the process of X-chromosome inactivation during which one of the two X chromosomes in the female genome has to be deactivated (in order to compensate for the fact that males only have one X chromosome and hence not overproduce proteins from both X chromosomes in females). This is done by “silencing” genes: changing the structure of the chromosome into more densely packed heterochromatin thereby deactivating whole regions of the genome.21 We suggest that chromatin consists of chromatin fibers that are either flexible and hence more “open”, or semiflexible and more compact. Therefore, the persistence length of chromatin fiber controls the densities of two possible chromosomal states. Toroidal morphologies that we observe in simulations (Figure 9) have been identified by biological and synthetic polymer experiments for numerous systems (for review see5) ranging from DNA packaging in the viral capsid to toroid-like organization of the genetic material in mammalian sperm cells.22 Other biopolymers such as actin (lp = 17 μm), or xanthan-chitosan complexes (lp = 120 nm) also have an ability to form toroid-like structures. For chains with smaller persistence lengths, such as alginate (lp = 10 nm), there is evidence of metastable structures that do not exhibit toroid morphology. Our simulations agree with experimental results,5 showing that metastable tori and toroid-like structures are only formed within a narrow range of persistence lengths. According to Maurstad et al.5 due to a very short time scale of the collapse process, high time resolution is required for experiments, which explains why the dynamics of collapse of biomolecules cannot be easily studied experimentally; e.g., DNA folding is considered to be too rapid to be able to follow the collapse process. However, experimentally, xanthan− chitosan complex collapse is a valuable general model of the coil−globule transition of semiflexible polymers. Experimental studies by Maurstad et al. show that these semiflexible polymers
Figure 9. Experimental data from the tapping-mode AFM height topograms of xanthan−chitosan compaction at room temperature (left), kindly provided by B. T. Stokke (partially from ref 5), is in qualitative agreement with numerical results presented in this work: there exist many deep metastable states with long lifetimes. Toroidal configurations are very stable in the window of lp ≈ 30 − 50, however, we also find many instances of chain collapse at these persistence lengths, even though some structures clearly belong to the same topological genus.
form toroidal and rod-like structures in their lowest energy (collapsed) state. This supports and complements our simulation results, which also show both toroid and rod-like states, as well as a collection of metastable intermediates. Hence, our work qualitatively agrees with experimental results found in the literature, 5 (Figure 9) and also demonstrates the dynamics of the collapse process in detail, which is crucial in understanding biological compactionassociated dynamic and morphological properties of polymer chains.
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DISCUSSION In this work, we show the dependency of the coil globule transition on chain stiffness. Using Brownian dynamics simulations, the process of collapse from its initial stages until the polymers have reached their global free energy minimum was followed in detail. We define monomer affinity in two different ways, counting particles within two specified radii 1.1σ and 2σ, which represent two types of contacts: bonds (of covalent and noncovalent nature) and the average local density, respectively. By examining contact density profiles of chains with different persistence lengths, we demonstrate that the optimal compaction density is reached at lp = 8σ; due to hairpin stacking, chains in the folded state at this persistence length are more compact than the folded flexible polymers. By analyzing the morphological properties of chains with the help of contact maps, we notice that the structures of all semiflexible polymers are based on hairpin loops, the size of which increases proportionally with stiffness. From the results of this work, it is clear that there are many metastable states of different morphologies, and hence racquet-like and toroidal shapes described in earlier theoretical work (ref 2) are probably only a fraction of the large ensemble of possible intermediate 7130
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(19) Terentjev, E. M.; Petschek, R. G. Phys. Rev. A 1992, 45, 930− 938. (20) Cook, P. R. Principles of Nuclear Structure and Function; WileyLiss: New York, 2001. (21) Turner, B. M. Chromatin and Gene Regulation; Blackwell Publishing: London, 2001. (22) Hud, N. V.; Vilfan, I. D. Annu. Rev. Biophys. Biomol. Struct. 2005, 34, 295−318.
morphological states that might choose to either undergo the coil-to-toroid or coil-to-rod transformations, or even experience a toroid to rod-like chain transfiguration, depending on the topological symmetry of the chain in a toroid-like configuration with more than one genus. In order to understand the dynamics of the folding process and how this depends on the persistence length of a chain, we studied how the time to collapse changes with increasing stiffness. The time to collapse is a highly nonlinear function of persistence length, with an over an order of magnitude change between flexible and stiff chains. Finally, we report an excellent qualitative agreement between our numerical results and experimental studies on xanthan− chitosan compaction process, which is important to our detailed understanding of the process of polymer chain collapse, and shows that our simulations can be used to model both synthetic and biological systems, most of which cannot be studied experimentally due to inefficient time resolution of experiments.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors would like to thank Bjorn T. Stokke and Gjertrud Maurstad for their experimental data. This work was funded by the Osk. Huttunen Foundation (Finland) and performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam. ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England.
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REFERENCES
(1) Lifshitz, E. M. Sov. Phys. JETP 1969, 28, 1280−1286. (2) Schnurr, B.; MacKintosh, F. C.; Williams, D. R. M. Europhys. Lett. 2000, 51, 279. (3) Pereira, G. G.; Williams, D. R. M. ANZIAM J. 2004, 45, C163− C173. (4) Lappala, A.; Terentjev, E. M. Macromolecules 2013, 46, 1239− 1247. (5) Maurstad, G.; Stokke, B. T. Curr. Opin. Colloid Interface Sci. 2005, 10, 16−21. (6) Liu, S.; Muthukumar, M. J. Chem. Phys. 2002, 116, 9975−9982. (7) Ben-Naim, E.; Grest, G. S.; Witten, T. A.; Baljon, A. R. C. Phys. Rev. E 1996, 53, 1816−1822. (8) Das, S.; Chakraborty, S. J. Chem. Phys. 2010, 133, 174904. (9) Cao, Q.; Zuo, C.; Ma, Y.; Li, L.; Chen, S.; Hu, Z. J. Polym. Sci., Part B: Polym. Phys. 2011, 49, 882−889. (10) Kolinski, A.; Skolnick, J.; Yaris, R. J. Chem. Phys. 1986, 85, 3585−3597. (11) Schnurr, B.; Gittes, F.; MacKintosh, F. C. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2002, 65, 061904. (12) Noguchi, H.; Yoshikawa, K. J. Chem. Phys. 2000, 113, 854−862. (13) Kremer, K.; Grest, G. S. J. Chem. Phys. 1990, 8, 5057−5086. (14) Auhl, R.; Everaers, R.; Grest, G. S.; Kremer, K.; Plimpton, S. J. J. Chem. Phys. 2003, 119, 12718−12728. (15) Rosa, A.; Everaers, R. PLoS Comput. Biol. 2008, 4, e1000153. (16) Liu, J. X.; J., D. T. Powder Metall. 1998, 53, 35. (17) Anderson, V. J.; Terentjev, E. M. Eur. Phys. J. E 2001, 4, 21−28. (18) Warner, M.; Gunn, J. M. F.; Baumgartner, A. J. Phys. A 1992, 18, 3007. 7131
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