# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{tot}= M.N. All semiflexible chains have the same sequence of bead types, S = {s

_{1},s

_{2},…,s

_{N}}, and all beads have the same diameter σ. Electrostatic interactions are represented by a screened Coulomb potential, hydrophobic attraction and excluded volume interactions by a Lennard–Jones potential. Hence, the non-bonded interaction between two beads i and j with bead types s

_{i}and s

_{j}is:

_{B}T. The Debye screening length is ${\mathsf{\kappa}}^{-1}={\left(8\mathsf{\pi}{l}_{B}{n}_{s}\right)}^{-1/2}$, where n

_{s}is the number density of (monovalent) ions. The effective bead charges ${q}_{{s}_{i}}$ (number of elementary charges e) are:

_{ijk}is the angle between the (i)…(j) and (j)…(k) virtual bond vectors. The linear chains are assumed to be straight at equilibrium hence the equilibrium bending angle is zero, and j = i + 1, k = i + 2.

_{p}of collagen triple helices cover a wide range of values [20,21].

_{p}≈ 35 nm which is on the lower side of the reported values. We do so since we would rather like to overestimate the effect of bending fluctuations on the fibril stability than to underestimate it. Collagen amino acid sequences are repeats of G-Xaa-Yaa triplets, where Xaa is often proline and Yaa often hydroxyproline. If we let each bead represent the helical rise 0.85 nm of a single G-Xaa-Yaa triplet of amino acids, we arrive at a bond distance l

_{b}= 0.57σ. A single interaction bead then represents 9 amino acids (3 strands, a G-Xaa-Yaa triplet on each strand). Given a persistence length l

_{p}≈ 35 nm, we then arrive at a bending elastic constant k

_{b}= 10k

_{B}T. We assume a typical Debye length of κ

^{−1}= 1 nm, or κσ = 1.5, corresponding to an equivalent concentration of monovalent electrolyte of approximately 0.1 M. Values for the effective charges of q of the positively and negatively charged beads, and for the Lennard–Jones energy parameter ${\u03f5}_{h}$ for hydrophobic beads are chosen such that the attraction between the beads (at a distance corresponding to the minimum of the Lennard–Jones potential) is approximately equal to the thermal energy k

_{B}T. The Lennard–Jones energy parameter ${\u03f5}_{0}$ for the charged beads and for the uncharged hydrophilic beads is chosen to be small such that the Lennard–Jones attraction between these beads is negligible as compared to the thermal energy k

_{B}T. Numerical values of all parameters are given in Table 1.

_{rod}(S) of a central chain with the surrounding chains in a staggered crystalline arrangement, we use a Metropolis Monte Carlo search algorithm. Sequences S are assumed to be palindromic. The initial sequence is a random (but palindromic) sequence with 30% hydrophobic, 30% uncharged hydrophilic and 40% charged beads (half of which are positively and half of which are negatively charged). Two random beads are selected in the first half of the sequence. Random selection is repeated until two beads are found of a different type. A Monte Carlo trial move consists of exchanging these two beads in both halves of the palindromic (symmetric) sequence, yielding a new sequence S

_{new}. If ΔE

_{rod}= E

_{rod}(S

_{new}) − E

_{rod}(S) < 0, the trial move is accepted. If ΔE

_{rod}> 0, it is accepted with probability exp(−ΔE

_{rod}/ΔE). We choose ΔE = 2k

_{B}T. All sequences S and corresponding interaction energies E

_{rod}(S) produced by the runs are stored for later analysis. To speed up the calculations, before the start of the minimalization, neighbor lists are prepared and used for each packing, for a cut-off interaction radius of the beads of r

_{cutoff}= 5σ.

_{0}and S

_{rand}

_{0}= {4,4,4,4,2,2,2,2,1,1,1,1,1,1,3,3,3,3,2,2,2,2,3,3,3,3,1,1,1,1,1,1,2,2,2,2,4,4,4,4}

_{rand}= {2,2,4,3,2,3,4,1,1,2,3,1,2,1,3,2,4,1,1,4,4,1,1,4,2,3,1,2,1,3,2,1,1,4,3,2,3,4,2,2}

_{r}of the rods, has physical meaning. For a rod consisting of N beads of diameter σ, spaced at a distance l

_{b}, in the absence of bead–bead hydrodynamic interactions, ${\tau}_{r}={\zeta}_{0}{N}^{3}{l}_{b}{}^{2}/12{k}_{B}T$, where ${\zeta}_{0}$ is the hydrodynamic friction of a single bead. For our parameters this gives τ

_{r}= 8.67 × 10

^{4}τ (at $\tilde{T}=1)$. The cut-off for both the Lennard–Jones and Debye–Hückel interactions was set to 5σ. Non-bonded interactions were only taken into account between beads on different chains. For assembly simulations a rectangular simulation box was used with dimensions (l

_{x},l

_{y},l

_{z}) = (20σ,20σ,120σ). Reflective rather than periodic boundary conditions were used in order to enforce the formation of finite rather than infinite fibrils. Initially, M = 40 rods were placed regularly in the box, with their long axis pointing in the z-direction. Particles were given random initial velocities corresponding to a LJ temperature $\tilde{T}=5$ and a simulation was run for 58τ

_{r}(or 10

^{8}simulation steps). Next, particles were given random initial velocities corresponding to a LJ temperature $\tilde{T}=2.5$ and a simulation was again run for 58τ

_{r}(or 10

^{8}simulation steps). Finally, particles were given random initial velocities corresponding to a LJ temperature $\tilde{T}=1.5$ and a simulation was run for 46τ

_{r}(or 8 × 10

^{7}simulation steps). During equilibration, temperatures for all Langevin dynamics runs, decreases somewhat. For the run with $\tilde{T}=1.5$, the final equilibrium temperature was $\tilde{T}=1.0$. By running the simulations for a large number of rotational diffusion times, we guarantee that thermal equilibrium can indeed be reached during the simulations.

_{0}are used that together form the non-gapped region. These are:

_{0,A}= {4,4,4,4,2,2,2,2,1,1,1,1}

_{0,B}= {3,3,3,3,2,2,2,2,3,3,3,3},

_{0,C}= {1,1,1,1,2,2,2,2,4,4,4,4}.

_{1}packing of the S

_{0}sequence, with a lattice spacing l

_{c}= 1.3σ. For these simulations, a rectangular simulation box was used with dimensions (l

_{x},l

_{y},l

_{z}) = (24σ,18σ,11.84σ). Again, reflective boundary conditions were used. Particles were given random initial velocities corresponding to an LJ temperature $\tilde{T}=1.4$. After equilibration, this leads to an average temperature during the simulations of $\tilde{T}=1$. The rotational diffusion time for these much shorter rods with N = 12 is τ

_{r}= 2.34 × 10

^{3}τ. Simulations were run for 21τ

_{r}(or 10

^{6}simulation steps).

## 3. Results

#### 3.1. Coarse-Grained Model for Patterned Semiflexible Chains

_{b}) with harmonic bond stretching energy and harmonic bending energy. Electrostatic interactions are represented using a screened Coulomb interaction potential, steric and hydrophobic interactions using a Lennard–Jones potential. We focus on sequence variation and therefore choose a single set of interaction parameter values representative for semiflexible collagen-like triple helices (see Section 2). Specifically, we assume a persistence length of the semiflexible chains of l

_{p}= 20σ.

_{1},s

_{2},…,s

_{N}}, where the bead type s

_{i}= 1…4. For the values of the non-bonded interaction parameters that we use (Table 1), contact interactions between the different types of beads are on the order of the thermal energy k

_{B}T, such that multiple attractive contact interactions along the chains will be necessary to stabilize the fibrillar state.

#### 3.2. Target-Staggered Crystalline Arrangements

_{rod}(S) of patterned rods arranged in staggered hexagonal lattices, with respect to their sequence S. If the ground state that is found in this manner is sufficiently deep, we anticipate it will be stable against the thermally excited shape fluctuations of the semiflexible chains. This will be checked explicitly by verifying that semiflexible chains with the optimized sequences do indeed assemble into the designed crystal structure, using Langevin Dynamics simulations. Similar ground state design approaches are very successful for designing, e.g., de-novo designed self-assembling proteins [2].

_{b}, aligned with their long axis in the z-direction, with a hexagonal arrangement in the x-y plane. The hexagonal lattice spacing is l

_{c}. In the x-z plane, neighbouring rods are shifted in the z-direction by the stagger D. The geometry of the family of lattices is illustrated in Figure 2. The ratio L/D determines both the periodicity ${N}_{\parallel}$ of the staggered arrangement and the gap width g:

**r**

_{ijk}of the origins of the rods in the lattices are:

_{b}. Furthermore, we restrict ourselves to values of the gap g, 2 ≤ g/l

_{b}≤ 8, and lattice periodicities up to ${N}_{\parallel}$ = 6. This allows for eight distinct staggered hexagonal crystalline packings of rods to be used in the numerical calculations P

_{1}–P

_{8}, and these are listed in Table 2. As an example, Figure 2 shows the three-dimensional structure of the P

_{5}packing.

#### 3.3. Sequence Optimization

_{1}–P

_{8}that we consider here, each chain has the same environment. Hence, as the objective function for the sequence design we can use the interaction energy E

_{rod}(S) of the central rod with lattice indices (i,j,k) = (0,0,0), with all the other chains in the crystalline arrangement:

_{n}and s

_{m}, and for a bead–bead distance ${r}_{nm}^{ijk}$ (see Section 2). For sequence optimization, we minimize the rod interaction energy E

_{rod}(S) with respect to the sequence S, for each of the packings P

_{1}–P

_{8}. We do so for a fixed composition in terms of bead types. Bead compositions we choose do not reflect natural collagen-like sequences, but rather reflect our expectations as to a typical bead composition that would readily form fibrils. We expect fibril assembly will be opposed by a net charge, while both charged and hydrophobic beads are necessary to drive assembly into staggered hexagonal packings. Too many hydrophobic beads could lead to kinetically trapped assembly into off-target packings. These considerations lead us to choose, for the study here, a composition of 30% of hydrophobic beads, 30% uncharged hydrophilic beads, and 40% charged beads (half of which are positively and half of which are negatively charged).

_{rod}(S) using a Metropolis Monte Carlo search algorithm that generates random sequences S with probability:

_{1}–P

_{8}we search for sequences S with minimal interaction energy E

_{rod}(S) in runs of 10

^{5}Monte Carlo steps (per packing). Results are shown in Figure 3. The lowest energy sequences are found for packings P

_{1}–P

_{3}with the lowest periodicity, ${N}_{\parallel}$ = 3. Differences in results between these lattices are very small. We choose the P

_{1}lattice for a more detailed investigation. Next, to get a complete histogram of the distribution of E

_{rod}(S) for all possible sequences S arranged in a P

_{1}lattice, we performed a simple random sampling search of 10

^{6}steps. The resulting energy distribution is shown in Figure 4a. For the bead compositions that we chose, we find that the most probable value for E

_{rod}is negative. Even random sequences with this bead type composition might therefore have a tendency to stick to each other and bundle up, albeit probably not in an ordered array. This we will check later using Langevin Dynamics simulations for a sequence S

_{rand}was randomly chosen from the set of sequences with interaction energy close to the peak of the energy distribution (given in Figure 5a).

^{6}steps is shown in Figure 4b, the 22 lowest energy sequences are illustrated in Figure 5a. The interaction energy is invariant with respect to exchange of the positive and negative beads, so in fact there are only 11 distinct minimum sequences for packing the chains into the P

_{1}packing, that we label in order of ascending rod interaction energy, S

_{0}–S

_{10}The striking feature of these minimum sequences, compared to the random sequence, is that all show clustering of similar bead types. The two lowest energy sequences, S

_{0}and S

_{1}, exhibit the most regular and simple patterns of stretches of bead types. The packing of chains with the sequence S

_{0}in the P

_{1}lattice is illustrated in Figure 5b,c.

_{0}–S

_{10}might correctly assemble into the designed P

_{1}structure also in the presence of thermal motion. A possible reason for exploring multiple sequences might be that the physical properties of P

_{1}fibrils, such as the mechanical properties, need not be the same. This, however, we leave as a topic for further research. Here, we restrict ourselves to establishing that designed sequences indeed assemble into the designed structures even if thermal motion is switched on. This we do for the lowest energy sequence S

_{0}.

#### 3.4. Langevin Dynamics Simulations

_{rod}(S) for a given target packing P. This “ground-state” approximation for sequence design is shown to be very powerful in other cases, such as the de-novo computational design of protein sequences [2], but it cannot guarantee that the target structure will be adopted by the optimized sequence. First of all, entropy competes with ordering and may preclude the ordered packing from being actually realized at finite temperatures. Furthermore, the optimized sequence may have yet lower rod interaction energies in arrangements other than the target packing. Finally, kinetic barriers may prevent the target structure from being adopted. Therefore, we check using Langevin Dynamics simulations, whether semiflexible chains with the optimized sequences indeed spontaneously assemble in the target packing or not. We do so for the S

_{0}sequence, which was computationally optimized to assemble into the P

_{1}structure, and compare with a typical random sequence, S

_{rand}. Random non-assembled initial states for M = 40 semiflexible chains were prepared at a temperature $\tilde{T}=1.4$ by stepwise cooling from still higher temperatures. A rectangular, closed simulation box was employed. To start assembly, the system was cooled to $\tilde{T}=1$, and simulated for t = 46τ

_{r}(or 8 × 10

^{7}simulation steps).

_{rand}we observe clustering of the chains into a very loosely associated anisotropic aggregate, but no clear fibril formation (Figure 6a). This is associated with just a small decrease in the total interaction energy E

_{int}per interaction bead (Figure 6c). On the other hand, for the S

_{0}sequence, we observe rapid staggered assembly into fibrils (Figure 6b), associated with a large decrease in the interaction energy E

_{int}per interaction bead (Figure 6c). A zoom of the final fibril structure for the optimized S

_{0}sequence is shown in Figure 6d. How close is this to the target P

_{1}packing it was designed to assemble into, and which is shown in Figure 5b,c?

_{1}target structure. For the cross-sectional order, in the plane perpendicular to the long axis of the fibril, it is more difficult to establish whether this matches the P

_{1}packing or not, because of the relatively low number of chains in the cross-sections. Representative cross-sections are shown in Figure 6d. The left cross-section, through the charged part of the chains, is expected to show the structure indicated in Figure 5c (top): positively charged beads should have a surrounding consisting of alternating negatively charged and neutral beads, etc. This can indeed be qualitatively recognized, but the numbers of chains in the cross-section are too low to be sure about the regularity of the pattern. Both this cross-section and the other cross-section, through the hydrophobic part of the chain, should show hexagonal order, but again, the numbers of chains in the cross-sections are too low to be absolutely sure.

_{0}sequence is indeed the hexagonal P

_{1}packing, we performed Langevin-Dynamics simulations for the subsequences of the S

_{0}sequence, which together form the non-gapped part of the fibril. These were arranged in a 10 × 10 hexagonal lattice, and Langevin-Dynamics at $\tilde{T}=1$ was used to establish the stability of this lattice against the thermal motion. Results are shown in Figure 7. The 10 × 10 hexagonal starting lattice is shown in Figure 6a. The final configuration, at t = 23τ

_{r}(or 10

^{6}simulation steps) is shown in Figure 7b. The top view shown in Figure 6c more clearly shows the ordering that remains in the presence of thermal fluctuations. The interaction energy per bead during the simulation run is shown in Figure 7d. During the simulation, the initial hexagonal lattice contracts somewhat, lowering the interaction energy per bead, and then remains constant. Especially from Figure 7d, it is evident that at $\tilde{T}=1$, thermal motion does not destroy the initial hexagonal ordering for these larger 10 × 10 cross-sectional lattices. Some disorder is observed at the edges, but not in the central part of the lattice. We conclude therefore that at to $\tilde{T}=1$, the S

_{0}sequence indeed spontaneously assembles into the target-staggered hexagonal P

_{1}structure, as designed.

## 4. Discussion

_{0}sequence can also be considered as a more general pattern for staggered assembly into hexagonal (quasi) crystalline fibrils, independent of the specific type of bead interactions. This is illustrated in Figure 8. The S

_{0}sequence is palindromic (non-polar) and consists of two elements: a polar α domain, and a non-polar β domain, as shown in Figure 8a,b. Favorable interactions for the domains are antiparallel interactions of α-domains, and interactions of β-domains with α domains in any orientation. Real systems such as helical peptides and proteins are polar. A possible domain structure for a generic polar interaction pattern for driving staggered assembly into a hexagonal lattice immediately follows from the non-polar case of Figure 8b, and is shown in Figure 8c. It is a sequence of three polar domains, α, β and γ. Staggered assembly into a hexagonal lattice will ensue if interactions of the domains are such that the individual parallel α, β and γ domains form a hexagonal sheet with a pattern of alternating α, β and γ domains. This will happen, for example, if the only attractive interactions are α–β, β–γ and γ–α.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Seeman, N.C.; Sleiman, H.F. DNA nanotechnology. Nat. Rev. Mater.
**2017**, 3, 1–23. [Google Scholar] - Huang, P.-S.; Boyken, S.E.; Baker, D. The coming of age of de novo protein design. Nat. Cell Biol.
**2016**, 537, 320–327. [Google Scholar] [CrossRef] - Torquato, S. Inverse optimization techniques for targeted self-assembly. Soft Matter
**2009**, 5, 1157–1173. [Google Scholar] [CrossRef] [Green Version] - Engel, M.; Damasceno, P.F.; Phillips, C.L.; Glotzer, S.C. Computational self-assembly of a one-component icosahedral quasicrystal. Nat. Mater.
**2015**, 14, 109–116. [Google Scholar] [CrossRef] - Wilber, A.W.; Doye, J.P.K.; Louis, A.A.; Noya, E.G.; Miller, M.A.; Wong, P. Reversible self-assembly of patchy particles into monodisperse icosahedral clusters. J. Chem. Phys.
**2007**, 127, 085106. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Romano, F.; Sciortino, F. Patterning symmetry in the rational design of colloidal crystals. Nat. Commun.
**2012**, 3, 975–976. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hulmes, D.J.S.; Miller, A.; Parry, D.A.D.; Piez, K.A.; Woodhead-Galloway, J. Analysis of the primary structure of collagen for the origins of molecular packing. J. Mol. Biol.
**1973**, 79, 137–148. [Google Scholar] [CrossRef] - Doyle, B.B.; Hukins, D.W.L.; Hulmes, D.J.S.; Miller, A.; Rattew, C.J.; Woodhead-Galloway, J. Origins and implications of the D stagger in collagen. Biochem. Biophys. Res. Commun.
**1974**, 60, 858–864. [Google Scholar] [CrossRef] - Meng, J.J.; Khan, S.; Ip, W. Charge interactions in the rod domain drive formation of tetramers during intermediate filament assembly. J. Biol. Chem.
**1994**, 269, 18679–18685. [Google Scholar] - Eikenberry, E.F.; Brodsky, B. X-ray diffraction of reconstituted collagen fibers. J. Mol. Biol.
**1980**, 144, 397–404. [Google Scholar] [CrossRef] - Kaur, P.J.; Strawn, R.; Bai, H.; Xu, K.; Ordas, G.; Matsui, H.; Xu, Y. The Self-assembly of a Mini-fibril with Axial Periodicity from a Designed Collagen-mimetic Triple Helix. J. Biol. Chem.
**2015**, 290, 9251–9261. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Strawn, R.; Chen, F.; Haven, P.J.; Wong, S.; Park-Arias, A.; De Leeuw, M.; Xu, Y. To achieve self-assembled collagen mimetic fibrils using designed peptides. Biopolymers
**2018**, 109, e23226. [Google Scholar] [CrossRef] [PubMed] - Chen, F.; Strawn, R.; Xu, Y. The predominant roles of the sequence periodicity in the self-assembly of collagen-mimetic mini-fibrils. Protein Sci.
**2019**, 28, 1640–1651. [Google Scholar] [CrossRef] [PubMed] - Kouwer, P.H.J.; Koepf, M.; Le Sage, V.A.A.; Jaspers, M.; Van Buul, A.M.; Eksteen-Akeroyd, Z.H.; Woltinge, T.; Schwartz, E.; Kitto, H.J.; Hoogenboom, R.; et al. Responsive biomimetic networks from polyisocyanopeptide hydrogels. Nat. Cell Biol.
**2013**, 493, 651–655. [Google Scholar] [CrossRef] [Green Version] - Włodarczyk-Biegun, M.K.; Slingerland, C.J.; Werten, M.W.T.; Van Hees, I.A.; De Wolf, F.A.; De Vries, R.; Stuart, M.A.C.; Kamperman, M. Heparin as a Bundler in a Self-Assembled Fibrous Network of Functionalized Protein-Based Polymers. Biomacromolecules
**2016**, 17, 2063–2072. [Google Scholar] [CrossRef] [PubMed] - Freeman, R.; Han, M.; Álvarez, Z.; Lewis, J.A.; Wester, J.R.; Stephanopoulos, N.; McClendon, M.T.; Lynsky, C.; Godbe, J.M.; Sangji, H.; et al. Reversible self-assembly of superstructured networks. Science
**2018**, 362, 808–813. [Google Scholar] [CrossRef] [Green Version] - Rele, S.; Song, Y.; Apkarian, R.P.; Qu, Z.; Conticello, V.P.; Chaikof, E.L. D-Periodic Collagen-Mimetic Microfibers. J. Am. Chem. Soc.
**2007**, 129, 14780–14787. [Google Scholar] [CrossRef] - Sharp, T.H.; Bruning, M.; Mantell, J.; Sessions, R.B.; Thomson, A.R.; Zaccai, N.R.; Brady, R.L.; Verkade, P.; Woolfson, D.N. Cryo-transmission electron microscopy structure of a gigadalton peptide fiber of de novo design. Proc. Natl. Acad. Sci. USA
**2012**, 109, 13266–13271. [Google Scholar] [CrossRef] [Green Version] - Rosenbaum, D.M.; Liu, D.R. Efficient and Sequence-Specific DNA-Templated Polymerization of Peptide Nucleic Acid Aldehydes. J. Am. Chem. Soc.
**2003**, 125, 13924–13925. [Google Scholar] [CrossRef] - Vesentini, S.; Redaelli, A.; Gautieri, A. Nanomechanics of collagen microfibrils. Muscles Ligaments Tendons J.
**2013**, 3, 23–34. [Google Scholar] [CrossRef] - Kirkness, M.W.; Lehmann, K.; Forde, N.R. Mechanics and structural stability of the collagen triple helix. Curr. Opin. Chem. Biol.
**2019**, 53, 98–105. [Google Scholar] [CrossRef] [PubMed] [Green Version] - LAMMPS Molecular Dynamics Simulator. Available online: https://lammps.sandia.gov/ (accessed on 1 September 2020).
- Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Grønbech-Jensen, N.; Farago, O. A simple and effective Verlet-type algorithm for simulating Langevin dynamics. Mol. Phys.
**2013**, 111, 983–991. [Google Scholar] [CrossRef] - Grønbech-Jensen, N.; Hayre, N.R.; Farago, O. Application of the G-JF discrete-time thermostat for fast and accurate molecular simulations. Comput. Phys. Commun.
**2014**, 185, 524–527. [Google Scholar] [CrossRef] [Green Version] - Ramshaw, J.A.M.; Shah, N.K.; Brodsky, B. Gly-X-Y Tripeptide Frequencies in Collagen: A Context for Host–Guest Triple-Helical Peptides. J. Struct. Biol.
**1998**, 122, 86–91. [Google Scholar] [CrossRef] - Bromley, E.H.C.; Channon, K.; Moutevelis, E.; Woolfson, D.N. Peptide and Protein Building Blocks for Synthetic Biology: From Programming Biomolecules to Self-Organized Biomolecular Systems. ACS Chem. Biol.
**2008**, 3, 38–50. [Google Scholar] [CrossRef] - Reinke, A.W.; Grant, R.A.; Keating, A.E. A Synthetic Coiled-Coil Interactome Provides Heterospecific Modules for Molecular Engineering. J. Am. Chem. Soc.
**2010**, 132, 6025–6031. [Google Scholar] [CrossRef] [Green Version] - Fletcher, J.M.; Harniman, R.L.; Barnes, F.R.H.; Boyle, A.L.; Collins, A.; Mantell, J.; Sharp, T.H.; Antognozzi, M.; Booth, P.J.; Linden, N.; et al. Self-Assembling Cages from Coiled-Coil Peptide Modules. Science
**2013**, 340, 595–599. [Google Scholar] [CrossRef] - Gradišar, H.; Božič, S.; Doles, T.; Vengust, D.; Hafner-Bratkovič, I.; Mertelj, A.; Webb, B.S.; Šali, A.; Klavžar, S.; Jerala, R. Design of a single-chain polypeptide tetrahedron assembled from coiled-coil segments. Nat. Chem. Biol.
**2013**, 9, 362–366. [Google Scholar] [CrossRef] [Green Version] - Locardi, E.; Kwak, J.; Scheraga, H.A.; Goodman, M. Thermodynamics of Formation of the Triple Helix from Free Chains and from Template-Constrained Chains of Collagen-like Monodisperse Poly(Gly-Pro-Hyp) Structures. J. Phys. Chem. A
**1999**, 103, 10561–10566. [Google Scholar] [CrossRef] - Persikov, A.V.; Ramshaw, J.A.M.; Brodsky, B. Prediction of Collagen Stability from Amino Acid Sequence. J. Biol. Chem.
**2005**, 280, 19343–19349. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Doig, A.J. Statistical Thermodynamics of the Collagen Triple-Helix/Coil Transition. Free Energies for Amino Acid Substitutions within the Triple-Helix. J. Phys. Chem. B
**2008**, 112, 15029–15033. [Google Scholar] [CrossRef] [PubMed] - Yu, Z.; Brodsky, B.; Inouye, M. Dissecting a Bacterial Collagen Domain fromStreptococcus pyogenes: Sequence and length-dependent variations in triple helix stability and folding. J. Biol. Chem.
**2011**, 286, 18960–18968. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yu, Z.; An, B.; Ramshaw, J.A.M.; Brodsky, B. Bacterial collagen-like proteins that form triple-helical structures. J. Struct. Biol.
**2014**, 186, 451–461. [Google Scholar] [CrossRef] [Green Version] - Alford, R.F.; Leaver-Fay, A.; Jeliazkov, J.R.; O’Meara, M.J.; DiMaio, F.P.; Park, H.; Shapovalov, M.V.; Renfrew, P.D.; Mulligan, V.K.; Kappel, K.; et al. The Rosetta All-Atom Energy Function for Macromolecular Modeling and Design. J. Chem. Theory Comput.
**2017**, 13, 3031–3048. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Illustration of coarse-graining of microscopic collagen triple-helical structure, to the coarse-grained representation in terms of interaction beads. Shown is the ideal triple helical structure for a (GPP)

_{4}-GPR-(GPP)

_{4}homotrimer, with three GPP triplets (one in each strand) being represented by a single hydrophilic (white) bead, and the three central GPR triplets (one in each strand) being represented by a positively charged (red) bead.

**Figure 2.**Representative packing chosen from the family of simple hexagonal staggered packings of (straight) chains considered. Shown is the packing P

_{5}, for (straight) chains of N = 40 interaction beads and length L = 40l

_{b}, arranged with a stagger D = 12l

_{b}such that the periodicity of the lattice is ${N}_{\parallel}$ = 4 and the gap is g = 8l

_{b}. (

**a**) Projection onto x-z plane, showing a single staggered layer of rods. The rod length L, stagger D and gap width g are indicated, as well as the crystal spacing l

_{c}and the bond length l

_{b}(separation between beads in the chains). (

**b**) Three-dimensional representation of cylindrical subsection of the crystalline lattice. Gapped and non-gapped regions of the fibril for which y-x cross-sections are shown in (

**c**), are indicated by (g) and (n). (

**c**) Projections on the x-y plane for non-gapped (n) and gapped region (g) indicated in (

**b**). Additionally indicated is the shift of 3/2l

_{c}in the x-direction, between successive layers of rods (in the x-z direction).

**Figure 3.**Lowest interaction energies E

_{rod,min}(units of k

_{B}T) found in Monte Carlo sequence optimization for chains of L = 40 beads (30% hydrophobic, 30% hydrophilic, 40% charged) arranged in staggered hexagonal packings P

_{1}–P

_{8}.

**Figure 4.**Sequence optimization for staggered hexagonal packing in P

_{1}lattice. (

**a**) Distribution of rod interaction energies E

_{rod}(units of k

_{B}T) from simple random sequence search (10

^{6}steps). A representative random sequence S

_{rand}is selected randomly from the sequences with energy for which the distribution peaks (vertical dotted line). (

**b**) Low energy tail of the distribution of rod interaction energies, obtained with Metropolis Monte Carlo sequence search (10

^{6}steps).

**Figure 5.**(

**a**) Lowest rod interaction energy sequences S

_{0}–S

_{10}from Metropolis Monte Carlo sequence search. Additionally shown is the representative random sequence S

_{rand}drawn from sequences with the rod interaction energy for which the probability distribution peaks, see Figure 4a. (

**b**,

**c**) Packing of optimized sequence S

_{0}into the staggered hexagonal P

_{1}target lattice. (

**b**) Cylindrical subsection of crystal lattice. (

**c**) slice in x-z plane: single layer of the lattice of chains, with staggered arrangement and perpendicular slices, in the x-y plane, demonstrating the three different perpendicular packings, one gapped and two non-gapped. Color code for interaction beads: green = hydrophobic (type 1), white = hydrophilic uncharged (type 2), red = positively charged (type 3), blue = negatively charged (type 4).

**Figure 6.**Langevin Dynamics simulation of assembly of chains with sequences S

_{0}and S

_{rand}after quench from $\tilde{T}$ = 1.4 to $\tilde{T}$ = 1. (

**a**) Initial (t = 0) and final (t = 75τ

_{r}) configurations for sequence S

_{rand}, where τ

_{r}is the rotational diffusion time of the corresponding rigid rods. (

**b**) Initial (t = 0) and final (t = 75τ

_{r}) configurations for sequence S

_{0}, where τ

_{r}is the rotational diffusion time of the corresponding rigid rods. (

**c**) Chain–chain interaction energy E

_{int}(per bead, in Lennard–Jones units, $\u03f5$) during the assembly, for chains with sequences S

_{rand}(red symbols) and S

_{0}(blue symbols), as a function of the simulation time in Lennard Jones units, t, scaled by the diffusional rotation time τ

_{r}. (

**d**) Zoom-in of the final structure for the chains with sequence S

_{0}, with cross sections at the two different non-gapped regions.

**Figure 7.**Langevin Dynamics simulation of hexagonally packed assembly of chains with sequences S

_{0,A}, S

_{0,B}, S

_{0,C}, which are the subsequences of S

_{0}that together make up the non-gapped regions of fibrils assembled from chains with the sequence S

_{0}. The simulation demonstrates that the hexagonal cross-sectional structure expected for the P

_{1}packing, is stable against thermal fluctuations for the S

_{0}sequence. (

**a**) 10 × 10 hexagonally packed initial configuration. (

**b**) relaxed final configuration at t = 190τ

_{r}. (

**c**) relaxed final configuration, top view, which more clearly shows that the hexagonal ordering is maintained in the presence of thermal fluctuations. (

**d**) Chain–chain interaction energy E

_{int}(Lennard Jones units, $\u03f5$) versus simulation time t scaled by rotational diffusion time τ

_{r}of the rods.

**Figure 8.**Generalization of the patterned chain. (

**a**) For chains with the S

_{0}sequence, which have the simple staggered P

_{1}lattice as their lowest interaction energy state, the sequence is composed of two polar α-domains and a non-polar β-domain, separated by inert gap sequences (g). (

**b**) The non-polar β-domain interacts with two of the polar α-domains, in the opposite orientation, forming a two-dimensional hexagonal lattice (

**c**) generalization to polar chains with α, β, γ domains putatively assembling into hexagonal staggered P

_{1}lattices if there is a preference of the α, β, and γ domains for assembly in a two-dimensional hexagonal lattice in which the α, β and γ domains alternate.

Parameter | Value | Unit |
---|---|---|

κ | 1.5 | σ^{−1} |

l_{B} | 0.47 | σ |

q | 2.0 | − |

${\u03f5}_{h}$ | 2.0 | ${k}_{B}T$ |

${\u03f5}_{0}$ | 0.1 | ${k}_{B}T$ |

l_{b} | 0.57 | σ |

k_{s} | 100 | ${k}_{B}T/{\sigma}^{2}$ |

k_{b} | 10 | ${k}_{B}T$ |

**Table 2.**Parameters for the target-staggered hexagonal packings P

_{1}–P

_{8}of (straight) chains considered in the numerical calculations.

Packing | L/l_{b} | D/l_{b} | ${\mathit{N}}_{\parallel}$ | g/l_{b} |
---|---|---|---|---|

P_{1} | 40 | 14 | 3 | 2 |

P_{2} | 40 | 15 | 3 | 5 |

P_{3} | 40 | 16 | 3 | 8 |

P_{4} | 40 | 11 | 4 | 4 |

P_{5} | 40 | 12 | 4 | 8 |

P_{6} | 40 | 9 | 5 | 5 |

P_{7} | 40 | 8 | 6 | 8 |

P_{8} | 40 | 7 | 6 | 2 |

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**MDPI and ACS Style**

Jongeling, A.; Svaneborg, C.; Vries, R.d.
Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains. *Symmetry* **2020**, *12*, 1926.
https://doi.org/10.3390/sym12111926

**AMA Style**

Jongeling A, Svaneborg C, Vries Rd.
Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains. *Symmetry*. 2020; 12(11):1926.
https://doi.org/10.3390/sym12111926

**Chicago/Turabian Style**

Jongeling, Arnoud, Carsten Svaneborg, and Renko de Vries.
2020. "Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains" *Symmetry* 12, no. 11: 1926.
https://doi.org/10.3390/sym12111926