Packing Defects Propagated During the Growth of Toroidal

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Packing Defects Propagated During the Growth of Toroidal Condensates by Semiflexible Polymer Chains Atreya Dey, and Govardhan Reddy J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b07600 • Publication Date (Web): 11 Sep 2017 Downloaded from http://pubs.acs.org on September 15, 2017

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The Journal of Physical Chemistry

Packing Defects Propagated During the Growth of Toroidal Condensates by Semiflexible Polymer Chains Atreya Dey and Govardhan Reddy∗ Solid State and Structural Chemistry Unit, Indian Institute of Science, Bengaluru, Karnataka, India 560012 E-mail: [email protected] Phone: +91-80-22933533. Fax: +91-80-23601310

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Abstract Deciphering the principles of DNA condensation is important to understand problems such as genome packing and DNA compaction for delivery in gene therapy. DNA molecules condense into toroids and spindles upon the addition of multivalent ions. Nucleation of a loop in the semiflexible DNA chain is critical for both the toroid and spindle formation. To understand the structural differences in the nucleated loop, which cause bifurcation in the condensation pathways leading to toroid or spindle formation, we performed molecular dynamics simulations using a coarse-grained bead-spring polymer model. We find that the formation of a toroid or a spindle is correlated with the orientation of the chain segments close to the loop closure in the nucleated loop. Simulations show that toroids grow in size, when spindles in solution interact with a pre-existing toroid and merge into it by spooling around the circumference of the toroid forming multimolecular toroidal condensates. The merging of spindles with toroids is facile, indicating that this should be the dominant pathway through which the toroids grow in size. The Steinhardt bond order parameter analysis of the toroid cross-section shows that the chains pack in a hexagonal fashion. In agreement with the experiments there are regions in the toroid with good hexagonal packing and also with considerable disorder. The disorder in packing is due to the defects, which are propagated during the growth of toroids. In addition to the well-known crossover defect, we have identified three other forms of defects, which perturb hexagonal packing. The new defects identified in the simulations are amenable to experimental verification.

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Introduction The condensation of DNA into toroids and spindles in the presence of multivalent cations is studied extensively 1,2 as it has implications in understanding polyelectrolyte condensation in poor solvents, 3–7 serves as a model system to understand high density DNA packing in viral capsids, 8,9 and has applications in gene therapy. 10 Polymers, which behave as self-avoiding random coils in good solvents undergo a collapse transition to compact globules when the solvent is made poor by either decreasing the temperature or by adding co-solvents. 11–13 However, semiflexible chains like DNA condense 14–17 into toroids and spindles when the solvent quality is tuned by adding polyvalent counterions such as spermidine3+ , cobalt hexammine3+ , spermine4+ , polyamines, etc. Experiments and theory revealed that DNA in the presence of polyvalent counterions condense into toroids and spindles due to the following properties: (1) high persistence length (≈ 50 nm), (2) excluded volume due to the chain diameter (≈ 20 nm), (3) surface energy 18–21 and (4) attraction between the chain segments due to the correlations between the polyvalent counterions condensed on the chain. 1,22,23 Theory and calculations 18,19,24,25 predict that toroids are the stable state for highly stiff chains, while spindles become the stable state with decrease in chain stiffness. However, in experiments 26–29 both toroids and spindles are routinely found to co-exist. Experiments 30,31 and simulations 18,32,33 show that for both the toroid and spindle formation, nucleation of a loop in the semiflexible chain is critical. The key structural features of the nucleated loop, which leads to the bifurcation in condensation pathways to a toroid or spindle are not clear. Insight into the structural features of the nucleated loop will be extremely useful in controlling the condensation pathways to achieve the desired toroid population and dimensions. Experiments 29 also show that with the increase in condensation time, the relative population of spindles decreases, whereas the population of toroids increases. It is not clear how the relative population of toroids and spindles changes with time. Multiple possibilities exist here, for example, spindles over a period of time can interconvert to toroids, or spindles can merge with the existing toroids in 3



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solution and can form multimolecular toroidal structures. The thermodynamic and kinetic feasibility of these spindle to toroid interconversion pathways needs to be investigated. Understanding the packing structure of DNA inside the toroids and the factors that influence it has direct implications in understanding the packing of DNA in viral capsids 8,9 and DNA compaction for gene delivery. 10 Experiments show that in the presence of spermine4+ , DNA collapses inside the viral capsid with local hexagonal packing. 34 The early X-ray diffraction studies 35,36 of DNA toroids showed evidence that DNA in the cross-section of a toroid is packed in an orderly fashion and speculated that it has hexagonal packing. The cryoelectron microscopy images from Hud and Downing 37 showed hexagonal packing of DNA in the toroids. The images further show alternating regions of defective and ideal hexagonal packing of DNA. A well-known topological defect of DNA packing in toroids is the crossover defect. 37,38 Ideal hexagonal packing in a toroid is possible only if the toroid is formed from concentric circles. In a toroid formed by a continuous chain like DNA, which has finite thickness, it is topologically not feasible for the chain to pack in concentric circles leading to defects in the hexagonal packing. Other than the well-characterized chain crossover defect, it is not clear whether any other defects exist, which can disrupt the hexagonal packing of the DNA chains in the toroids. In this work we study the growth mechanism of DNA toroids, and chain packing defects in the toroid cross-section using a coarse-grained bead-spring model for the semiflexible polymer chain and molecular dynamics simulations. We studied the structure of loops nucleated in the chain to identify the features responsible for the bifurcation in the chain condensation pathways leading to either toroids or spindles. We find that the orientation of the chain segments near the loop closure play a critical role in determining whether the nucleated loop condenses into a toroid or spindle. The toroid and spindle structures did not inter-convert into one another on the time scale of the simulations. When spindles interact with a preexisting toroid in solution, they merged into the toroid by spooling around the circumference of the toroid leading to the formation of bigger multimolecular toroids. The merging of

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spindles with toroids is facile and happens on the time scale of the simulations indicating that this should be the dominant pathway through which the toroids grow in size. This also explains the experimental observation 29 of relative decrease in spindle population compared to toroids with the increase in condensation time. Metastable structures are observed during the condensation of thicker spindles onto toroids. Thicker spindles cannot easily spool around the circumference of a toroid. They have to partially unfold to successfully merge into the toroids and during the course of the unfolding transition, metastable structures are observed. To quantify the hexagonal packing of chains in the toroid cross-section, we have used the Steinhardt bond order parameters. 39,40 In agreement with the experiments 37 we have found regions in the toroid where there is ideal hexagonal packing as well as regions with disorder in the chain packing. The disorder in the chain packing is due to the defects, which are propagated during the growth of the toroids. In addition to the crossover defect, 37,38 we have found three other forms of defects, which were not identified before. These defects could be verified by experiments and the insights will help in building models to analyze the cryoelectron microscopy images of DNA toroids.

Model and Simulation Methods The contour length of DNA used in the experiments is on the order of micrometers. To bridge the length scale between experiments and simulations we used a coarse-grained beadspring model for the polymer chain representing DNA 41 in the simulations. The diameter of each bead in the chain is set to σ = 3.18 nm, which is equal to the pitch of DNA (10 base pairs). All the chains in the simulation have N = 256 beads, which is equivalent to DNA strands of length 2560 base pairs. In the chain, each bead is attached to its adjacent beads with a spring, which is modeled using a finite extensible non-linear elastic (FENE) potential given by VF EN E = −

NB X kR2 0

i=1

2

"

ln 1 −

5

ri − σ R0


2 #

,

(1)


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where NB (= 255) is the total number of bonds in the chain, ri is the distance between the centers of the ith pair of bonded beads, k (= 30 kB T /σ 2 ) is the spring constant, R0 (= 1.2 σ) is the maximum extension of the spring, kB is the Boltzmann constant, and T is the temperature. These parameters prevent intercrossing of the chains in the simulation. The persistence length of the chain is modeled using a harmonic bending potential given by Nang

Vbend =

X

α θi2 ,

(2)

i=1

where Nang (= 254) is the number of angles in the chain, θi is the deviation of the ith bond angle in the chain from 180◦ , and α (= 7.775 kB T /rad2 ) is a constant. For the value of α used in the simulations the persistence length of the chain is ≈ 50 nm, which is equivalent to the double stranded DNA persistence length. 42 The solvent quality on the polymer chain is taken into account implicitly using a truncated and shifted form of the Lennard-Jones (LJ) potential, which acts on the beads in the chain. To simulate the chain behavior in good solvent conditions a purely repulsive form of the LJ potential known as the Weeks-Chandler-Andersen (WCA) potential 43 is used and is given by

VW CA (r) =

 h   4ǫLJ   0

σ 12 r

−

σ 6 r

1 6

− c(2 σ)

i

1

r 6 26 σ

(3)

1 6

r > 2 σ,

where r is the distance between the two beads, ǫLJ is the attractive well depth, and the function c(r) is chosen such that the value of the potential is zero at the cut-off, i.e. c(r) = (σ/r)12 − (σ/r)6 . To simulate the behavior of the chain in poor solvent conditions, LJ potential with an attractive well-depth is used and is given by

VLJ =

   4ǫ   0

LJ

h

σ 12 r

−

σ 6 r 6

− c(2.5σ)

i


r 6 2.5σ r > 2.5σ.

(4)

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In good and poor solvent conditions, ǫLJ is set to 1.0 kB T and 1.2 kB T , respectively. 44 Individual single chains are initially equilibrated in good solvent conditions to form coil-like conformations. The collapse or condensation of a coil-like chain is triggered by changing the solvent conditions from good to poor. The DNA chain in poor solvents exist in one of these forms: toroid, spindle or coil-like conformations. The toroids and spindles nucleate in the solution from coil-like conformations. Experiments show that the toroids can contain multiple DNA chains. 1,16,31 This implies that the toroids grow in size when other chains in solution interact with the toroid and merge into it contributing to its growth. We simulated the growth of toroids, by tagging a chain in any conformation (toroid, spindle or coil-like) to a pre-existing toroid and simulated the merging of the chain into the toroid to form multimolecular toroids. To avoid the dead time of chain binding to the toroid in the simulations, we tagged the DNA chain to the toroid at a random position on the toroid. To prevent the tagged chain from getting detached we tethered the chain to the toroid using a harmonic potential given by Vteth =

k (r − σ)2 , 2

(5)

and the value of k is identical to the value used in the FENE potential (eq. 1). The dynamics of chain collapse and toroid growth in poor solvents is studied using Brownian dynamics simulations ignoring the hydrodynamic interactions. The equations of motion for the polymer beads are integrated by using the Ermack and McCammon 45 algorithm given by ri (t + ∆t) = ri (t) +

D ◦ Fi ∆t + Ri (∆t), kB T

(6)

where ri (t) is the position of bead i at time t, Fi is the deterministic force on bead i at time t, Ri (∆t) is a random displacement imparted to bead i sampled from a Gaussian distribution with mean zero and variance hRi (∆t) · Rj (∆t)i = 2D◦ ∆tδij , and D◦ is the single bead diffusion constant. The quantities σ, D◦ , and kB T are set to 1. Time is measured in units of τBD (= σ 2 /D◦ ), and the time-step used to advance the simulation is ∆t = 0.00012 τBD .

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Data Analysis: Radius of gyration, Rg , is used to characterize the size of the polymer chains and is defined as Rg =

N 1 X |ri − rCOM |2 N i=1

!1/2

,

(7)

where N is the number of beads in the polymer, ri and rCOM are the position vectors of bead i and center of mass of the chain, respectively. In the simulations of multimolecular toroid growth, which are initiated by the collapse of a coil-like chain or spindle onto a pre-existing toroid, we defined a parameter, R, to quantify the extent of collapse of the incoming chain and to probe whether it is deposited inside or outside the pre-existing toroid. The quantity R is defined as 1/2 N (j) X 1 (j) (1) R =  (j) |ri − rCOM |2  , N i=1 

(8)

where the superscript j varies between 1 and 2, and refers to the pre-existing toroid and (j)

incoming chain, respectively. ri

(1)

refers to the coordinates of bead i in molecule j. rCOM

is the center of mass of the pre-existing toroid. N (j) is the number of beads in molecule j. When R is calculated for j = 1, it is equal to the Rg of the pre-existing toroid. If the value of R for the incoming chain is greater than the Rg of the pre-existing toroid, then on an average, the incoming chain is deposited on the outer side of the toroid compared to the inner side close to the hole in the center of the toroid. To analyze the packing of chains in the cross-section of a toroid, we used the Steinhardt bond order parameters. 39,40 The cross-section of a toroid is processed as follows before computing the parameters. The procedure is illustrated in the schematic shown in Figure 1. A plane in which the toroid lies is iteratively fitted to the toroid using MATLAB. 46 A circle is drawn on the plane with the centre of mass of the toroid as centre and radius, r, equal to the average distance of the beads from the centre. To draw the circle, a vector a in the plane originating from the center of the toroid and length equal to r is chosen. A second

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vector b originating from the toroid center and length equal to r is obtained by taking the cross-product of vector a with the normal to the plane n (Figure 1). 1000 equidistant points on the circle are generated using the equation a cos(θ) + b sin(θ), and by varying the parameter θ from 0 to 2π. The resultant circle from these 1000 points will pass through the toroid and has a radius equal to r. At each of these 1000 points, planes of the toroid cross-section whose normals are the tangents to the circle at those points are analyzed for chain packing using the Steinhardt parameters (Figure 1). To identify the coordinates of the points where the chain passed through a plane, the coarse-grained bead-spring chain is converted into a continuous structure by linear interpolation. The coordinates where the continuous chain passed through each plane are analyzed using the Steinhardt bond order parameters for hexagonal packing 39,40 (Figure 1). The Steinhardt parameters are extensively used to quantify structural order present between particles. 47 These parameters quantify structure by assigning a unique value to a cluster of particles irrespective of their spatial orientation. The positions of particles in the clusters are projected onto the surface of a sphere. Discrete spherical harmonics transformation is used to represent the pattern on the surface of the sphere. The spherical coefficients of the transformation are complex vectors, ql , which are of 2l + 1 dimensions. The Steinhardt parameters, ql and wl , are the magnitudes of the complex vector ql computed using the equations 9 and 10, respectively. These parameters are rotationally invariant quantities and are unique to a cluster shape. To characterize the structure in the toroid cross-section, we computed the Steinhardt parameters, ql (i) and wl (i) for a point i in the cross-section using the equations v u l u 4π X t ql (i) = |qlm (i)|2 2l + 1 m=−l

10


(9)

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



l l   l  qlm1 (i)qlm2 (i)qlm3 (i)  m1 +m2 +m3 =0 m1 m2 m3 . wl (i) = l 3/2 P |qlm (i)|2 1 P

(10)

m=−l

In the above equations, qlm (i) is a complex vector defined as Nb (i) 1 X Ylm (rij ), qlm (i) = Nb (i) j=1

(11)

where Nb (i) is the number of nearest neighbors of point i in the toroid cross-section where the chain crosses it, and rij is the position vector from point j to i. Two points are considered nearest neighbors if the distance between them is less than 1.2σ. Ylm are the spherical harmonic functions, l is a free integer parameter, which is chosen to be 4 and 6, and m is an integer varying from −l to +l. The terms in the brackets in the numerator of eq 10 are the Wigner 3 − j symbols. 48 Table 1: Steinhardt bond order parameter values for ideal hexagonal packing 49 Parameters Value oB q4 0.375 oB q6 0.740829 oB w4 0.134097 w6oB -0.0462606 q4oS 0.440835 oS q6 0.759623 oS w4 0.125057 w6oS -0.0559095

To quantify the deviation of chain packing from hexagonal packing in the toroid crosssection, we computed the root mean square deviation (RMSD) of the ql (i) and wl (i) parameters from the hexagonal close packing values corresponding to the 111 plane (Table 1). Since the ql (i) and wl (i) values for the points on the surface differ from the values for the

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Results and Discussion Condensation of a Semiflexible Polymer Chain in Poor Solvent: To study the semiflexible polymer chain condensation in poor solvent, 100 independent simulations are spawned using different initial chain conformations extracted from the simulations performed in good solvent conditions. On changing the solvent conditions from good to poor, the chains mimicking the dsDNA condensed into toroids and spindles in agreement with the previous experiments 1,14,16,17,28,50–53 and simulations 18,19,24,32,33,44,54–60 (Figure 2). Out of the 100 independent simulations spawned, the chain at the end of the simulation time 70000 τBD , condensed into spindles (58%), toroids (41%) and coil-like conformations (1%). The energy and Rg of the collapsing chains as a function of time shows that toroids have the lowest energy and size compared to spindles 19,32,33 (Figure 2B, C). Among the spindles, structures with different number of racquete heads (hair-pin kind of loops) are observed (Figure 2A). The energy and size of the spindles decreases with the increase in the number of racquete heads, which help in packing the chain compactly in the condensed state 33 (Figure 2). On the time scale of the simulations (70000 τBD ) we did not observe any structural transition from spindles to toroids. However, from previous simulation studies, 33,44 there is evidence that partially formed spindles, with lower racquete head numbers can convert into toroids. The barriers for a fully condensed spindle to convert into a toroid will be high as it has to substantially unfold in poor solvent conditions to convert into a toroid, and unfolding in poor solvent conditions is highly unfavorable. For both the toroid and spindle formation, a loop has to nucleate in the chain. At the end of the loop, overhanging tails are present (Figure 2A, 3A). Subtle differences in the interactions between the two overhanging tails near the loop closure determines whether a chain forms a spindle or a toroid. We observed that the contacts between the beads present in the overhanging tails near the loop closure for toroids and spindles have positive and negative slopes, respectively in the contact map (Figure 3A). In a toroid nucleus, the DNA goes through a complete 360-degree loop (Figure 3A). As a result, near the loop closure, 13



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the bead numbers of the beads in contact from the two overhanging tails increase parallely, and the contacts give rise to a positive slope in the contact map. On the other hand, the DNA in the spindle nucleus goes through a 180-degree loop (Figure 3A). In the region where the beads from the overhanging tails are in contact, the bead numbers on one of the overhanging tails increases, and on the other tail the bead numbers decreases, leading to a negative slope by the contacts in the contact map. We hypothesized that the orientation of the overhanging tails with respect to each other near the loop closure should play a critical role in determining whether a nucleated loop forms a toroid or spindle. The orientation of the tails can strongly influence whether the nascent contacts near the loop closure will lead to contacts with a positive or negative slope (Figure 3A). To test this hypothesis, a loop consisting of 60 beads is created in the chain by tethering the bead numbers, n1 = 97, and, n2 = 157, using a harmonic potential to prevent the loop from falling apart. The polymer chain with the loop is then equilibrated in a good solvent using Brownian dynamics simulation. From the simulation trajectory we extracted 30,000 polymer conformations with the loop. For each conformation, the dihedral angle, φ, formed by the beads n1 − 5, n1 , n2 and n2 + 5 is calculated to quantify the orientation of the tails with respect to each other near the loop closure. The variation in φ from −π to π is divided into 20 bins of equal width, and each polymer conformation is assigned to one of these bins depending on its φ angle. From each of these 20 bins, 30 polymer conformations are chosen at random and using each conformation as the initial conformation, 5 short Brownian dynamics simulations are spawned in poor solvent conditions. The harmonic bond between n1 and n2 is turned off to allow for loop opening. A total of 3000 simulations are spawned. The final polymer conformations obtained from the simulations at the end of 700 τBD are analyzed for toroid or spindle nucleation using the slope of the contacts present between the polymer beads near the loop closure in the contact map. The contacts with positive and negative slopes are toroids and spindles, respectively (Figure 3A). Trajectories where the structure remains as a coil or if an additional loop nucleates in the chain are excluded from the analysis. The ratio of the

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number of spindles and toroids formed with respect to the total number of trajectories is plotted in a bar graph with 10 bins of equal width. The distribution clearly shows that when the angle φ ≈ 0, the structures tend to form spindles, and if φ ≈ −π or π, the structures go on to form toroids. This establishes that orientation of the tails near the loop closure plays an important role in determining the fate of the loop, i.e. whether the chain condenses into a spindle or toroid (Figure 3B). After the nucleation of a toroid in the chain, it grows further via a spooling mechanism, where the overhanging DNA tails continuously wrap around the nucleated toroid (Figure 4). The direction of spooling of the tails is however not unidirectional. The overhanging DNA tails can change spooling directions forming a “bridge” like structure along the outer wall of the toroid and these bridges are stable (Figure 4A). If a bridge is made of 9 beads for 2 a 180◦ turn in spooling direction, the bending energy cost is ≈ 9 · α · π9 = 8.53 kB T (α

value from eq 2). However, each of these 9 beads interact with at least 9 other beads on the surface of the toroid with an LJ interaction strength ǫLJ = 1.2 kB T . This attractive energy can compensate for the chain bending energy penalty in the bridge formation. Thus, shorter bridges can form if the value of ǫLJ is increased further. This implies that the probability of bridge formation is higher in poorer solvents (higher salt concentrations). Growth Mechanism of Toroids: Experiments show that in solutions of high DNA concentration, the toroids formed are made up of multiple chains. 1,16,22,31,37,61,62 The cryoelectron microscope images reported by Lambert et al. 62 show a single toroid being formed from the DNA of at least 10 T5 bacteriophages. Experiments 29 also show that the relative population of spindles decreases with the condensation time, whereas the population of toroids increases. This implies that spindles interconvert into toroids with time and the mechanism is not completely clear. There are at least three possible scenarios through which the spindles can convert into toroids: (1) Spindles undergo complete decondensation into coils and then renucleate to form toroids. (2) Spindles undergo internal rearrangement to form toroids. (3) Spindles interact with the toroids in solution, and merge with the toroids to form multi15




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molecular toroids. Both the scenarios (1) and (2) are not observed on the time scale of the simulations (70000 τBD ) performed in this work. There is a significant energy penalty for a condensed chain to unfold. But when we allow the interaction of spindles with toroids, on the time scale of the simulations, spindles merge with the toroids leading to multimolecular toroids. This shows that scenario (3) should be the dominant mechanism through which multimolecular toroids can grow. We studied the interaction of toroids and spindles with different number of racquete heads to understand how they merge to form multimolecular toroids (Figure 4). The dynamics of a coil-like chain (0 racquete heads) merging into a toroid is similar to the spooling mechanism through which a coil-like chain condenses into a toroid after the nucleation of a loop in the chain (Figure 4B). The 2 racquete head spindles also merge into the toroid via the spooling mechanism in a facile manner to form a toroid with 2 chains (Figure 4C, D). The quantity, R (defined in the Data Analysis section), for the chain merging into the toroid as a function of time shows that for all the trajectories, the R value steadily decay’s to the equilibrium value (Figure 4D). This implies that the collapse of a spindle with 2 racquete heads onto a toroid is facile since the nucleation step involving the loop formation is absent in this process. However, the timescales for the merging of spindles with racquete heads greater than 4 increases as these spindles are thicker in size and they cannot easily spool around the toroid. These spindles have to partially unfold to merge into the toroid. The partial unfolding and merging events are seen as transitions in R as a function of time plot (Figure 4D). A metastable state where the spindle has partially merged into the toroid is shown in Figure 4D. Similar looking metastable structures are also observed in the AFM and Cryo-TEM images taken during the intermediate stages of DNA condensation. 61,63,64 When a toroid in solution interacts with another toroid, their merging into a single toroid is not facile and they tend to get stuck in metastable T-shaped, non-toroidal states (Figure 4E). These T-shaped structures appear when one toroid goes sidewise into the hole of another toroid. These structures are also observed in the electron microscope images

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reported by Widom and Baldwin. 17 However, in some trajectories when the two toroids approach each other from one above the other, the final structure obtained after equilibration showed a toroidal like shape but the two DNA strands did not merge into one another on the time scale of the simulation (Figure 4E). This shows that the barriers for two toroids merging into a bigger toroid is high. Spooling like mechanism is also not observed during the merging of two toroidal structures as this would require at least one of the chains to unfold, which would incur a high energy penalty.

7.0

Toroid thickness

6.5

Radius ( )

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Toroid radius

6.0

1 chain 2 chains 3 chains 4 chains 5 chains

5.5

5.0 2

3

4

5

6

Thickness ( ) Figure 5: The inset shows the definition of radius and thickness of the cross-section of a toroid. The radius of toroids is plotted as a function of their thickness. Toroids with different number of chain molecules are colored differently. The linear fit through all the points show that radius increases with thickness. The increase in the radius with the number of chains in the toroid indicates that the chains are asymmetrically deposited on the outer surface of the toroid compared to the inner surface close to the center of the toroid. A plot of the quantity R of the incoming chain and the pre-existing toroid as a function of time shows that the equilibrium R values of the incoming chain are higher than the R values 19



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of the initial toroid (Figure 4D). This implies that a significant part of the incoming chain is deposited on the outer side of the toroid irrespective of the starting conditions, which is also evident from the simulation snapshots (Figure 4B,C). The asymmetric deposition of the incoming chain on the toroid can be due to the fact that the bending energy penalty is less when the chain deposits on the outer side of the toroid compared to the scenario where it deposits on the inner side closer to the hole in the toroid. If the asymmetric deposition of the incoming chains holds for the deposition of multiple chains on a toroid, then the radius of the toroid should increase with the thickness (see inset in Figure 5). As described in the methods section, we simulated the sequential collapse of 6 chains in different conformations (coil-like, and spindles with different number of heads) onto a single toroid. The radius of the toroid increased with thickness on adding multiple chains to a toroid (Figure 5). Preferential deposition of DNA on the outer side of the toroids is also observed in experiments 31 when the concentration of salts such as NaCl and MgCl2 used to control the ionic strength of solutions exceeded 2 mM. The coil-like chain spooling around the toroid showed occasional bridge formation by changing the spooling direction, and these bridges are generally found on the outer surface of the toroids (Figure 4B). When a spindle collapses onto the toroid, bridges are naturally observed in the final toroid as the chain in a spindle like conformation has at least one bridge in its conformation (Figure 4C). The presence of bridges in the toroid can lead to chain packing defects in the toroid cross-section. Packing Defects in Toroids: Cryoelectron microscopy images 2,37 of the toroids surface show well defined fringes of DNA along the circumference of the toroid. However, experiments 37 also point out that for a majority of the toroids well defined fringes are observed only for about one-third of the toroid circumference. Experiments 35,37 show evidence that the DNA inside a toroid has a patchy hexagonal packing. To identify the packing structure of the chains inside the toroids we computed the Steinhardt bond order parameters 39,40 for the cross section of the toroids as described in the methods section. To quantify the devi20


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ation of chain packing from hexagonal packing in the toroid cross-section we computed the RMSD from hexagonal packing using eq 12. The RMSD from hexagonal packing computed for various cross-sections in a toroid containing multiple chains is shown as a doughnut plot along with the actual toroid structure (Figure 6). The RMSD in the doughnut plot clearly shows regions in the toroid with good hexagonal packing, and also regions where packing significantly deviates from hexagonal packing (Figure 6A). A visual inspection of the toroid surface containing multiple chains shows that in the regions where the RMSD value is lower, the chains in the toroid appear to be packed in an orderly fashion, and in the regions where the RMSD value is higher, the disorder in the chain packing is high due to the defects in the packing (Figure 6B). Ideal hexagonal packing in the toroid is possible only if the toroid is formed by concentric circles (Figure 7A). In the toroid formed by the condensation of a long polymer chain, topologically it is not feasible for the chain to pack in concentric circles as it has finite thickness 37,38 (Figure 7B). This leads to a well-known defect know as the crossover defect, and it is the main cause of disruption from the ideal hexagonal packing. 37,38,65 In addition to the crossover defect, simulations show that there can be other defects due to the bridge formation. A bridge on the surface of the toroid is formed because of the change in the spooling direction of the overhanging part of the chain during toroid condensation (Figure 4A, B). A bridge also naturally forms when a spindle collapses onto a toroid (Figure 4C). Since the hair-pin like bridge is oriented perpendicular to the spooling direction it perturbs the local hexagonal packing of the DNA in the toroid cross-section. The existence of a bridge on the toroid surface causes other defects in the chain packing. When a chain spooling on the toroid surface encounters a bridge it leads to 3 different kinds of defects. In the first defect, the incoming chain is forced to spool over the bridge (Figure 7C). Spooling over the bridge is energetically unfavorable as it prevents the formation of optimal contacts between the chain monomers and introduces gaps in the packing. In the second defect, when the spooling chain encounters a bridge, it reverses its spooling direction forming

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Figure 7: (A) Ideal hexagonal packing in a toroid is possible only for concentric rings. (B) Cross-section of a toroid obtained from simulations. Hexagonal packing of the chains in a cross-section is highlighted in green. A part of the chain highlighted in red shows the crossover defect. Figures (C)-(E) show defects in the chain packing due to a hair-pin like bridge structure on the surface of the toroid. A bridge on the toroid surface is highlighted using blue colored beads. The chain with red colored beads on encountering a bridge on the toroid surface can (C) spool over the bridge, (D) reverse its spooling direction and form another bridge, (E) dislocate the bridge and spool past the bridge by making contacts with it.

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another bridge to avoid going over the pre-existing bridge (Figure 7D). In the third defect, the chain spools past the bridge by making contacts with the bridge and is distorted (Figure 7E). The identification of these new defects do not negate the possibility that crossovers defects are a major cause of perturbation in hexagonal packing. The simulations illuminate other sources of possible defects in the packing of semiflexible chains as toroids, which can be verified by experiments. These new insights on defects could be helpful in building new models to analyze the microscopy images of DNA toroids. The probability of observing defects related to the bridge formation in toroids should also depend on the sequence of DNA chain as it is well known that the local rigidity of the chain depends on the sequence. 66,67 DNA sequences rich in CC/GG dinucleotides are less rigid with a lower persistence length (≈ 42 nm) compared to sequences rich in AC/GT, CG or GA/TC dinucleotides, which are more rigid with a higher persistence length (≈ 55 nm). 67 Toroids formed from DNA sequences rich in CC/GG dinucleotides should have more defects related to the bridge formation as these sequences are less rigid and can easily form bridges.

Concluding Remarks To conclude we addressed problems related to the nucleation and growth of toroids by the condensation of semiflexible polymer chains in poor solvents using molecular dynamics simulations. We made the following testable predictions, which can be verified by experiments: (1) By controlling the orientation of chain segments near the loop closure, spindle formation can be suppressed and the nucleated loops in the polymer chain can be directed to condense into toroids. (2) The toroids grow in size by acquiring spindles from solution. Spindles interact and merge into the toroids by spooling around the circumference of the toroid contributing to the growth of toroids. (3) The chains in the toroid cross-section can have ideal hexagonal packing or disorder in packing. The disorder in the chain packing is due to the defects propagated during the growth of the toroids. In addition to the crossover defect we

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have identified 3 other defects due to the bridge formation on the surface of the toroid caused by the change in spooling direction of the chain during the toroid growth. We can speculate that the defects could have functional role during the decondensation of the compact DNA structures. For example in gene therapy, the compact DNA transferred into the cell should unfold to perform its intended function. The unfolding of the compact DNA could nucleate from the defects as the contacts between the DNA segments in the vicinity of defects are not optimal. Experiments further show that the structure of DNA can change from B-form to either C-form 65 or Z-form 68 upon condensation into toroids, and the structural changes depend on the sequence of DNA. Studying such structural changes in DNA upon condensation into toroids is beyond the scope of the coarse-grained DNA model we have used in the present study as it is devoid of any nucleotide identity.

Acknowledgement A part of this work is funded by the grants to G.R. from Science and Engineering Research Board (EMR/2016/001356) and Nano mission, Department of Science and Technology, India. A.D. acknowledges research fellowship from Indian Institute of Science-Bangalore. The computations are performed using the Cray XC40 cluster at IISc, PARAM Yuva-II and BRAF resources at C-DAC.

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Packing Defects Propagated During the Growth of Toroidal

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