# Interaction Patterns for Staggered Assembly of Fibrils from Semiflexible Chains
## 2 Materials and Methods
Given a persistence length \(l_{p}\approx 35\) nm, we then arrive at a bending elastic constant \(k_{b}=10k_{B}T\). We assume a typical Debye length of \(\kappa^{-1}=1\) nm, or \(\kappa\sigma=1.5\), 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 \(\epsilon_{h}\) 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 \(\epsilon_{0}\) 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.  
For energy minimalization of the total interaction energy \(E_{rad}(S)\) 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 \(\Delta E_{rad}=E_{rad}(S_{new})-E_{rad}(S)<0\), the trial move is accepted. If \(\Delta E_{rad}>0\), it is accepted with probability \(\exp(-\Delta E_{rad}/\Delta E)\). We choose \(\Delta E=2k_{B}T\). All sequences \(S\) and corresponding interaction energies \(E_{rad}(S)\) 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 \(r_{cutoff}=5\sigma\). Langevin Dynamics simulations of assembly into fibrils of optimized and a reference random sequences \(S_{0}\) and \(S_{rand}\)  
\(S_{0}=\{4,4,4,4,2,2,2,2,1,1,1,1,1,3,3,3,3,2,2,2,2,3,3,3,3,1,1,1,1,1,2,2,2,4,4,4\}\)  
\(S_{rand}=\{2,2,4,3,2,3,4,1,1,2,3,1,2,1,3,2,4,1,1,4,4,1,1,4,2,3,1,2,1,3,2,1,1,4,3,2,3,4,2,2\}\)  
(each having \(N=40\) beads) were performed using the LAMMPS software package [22; 23] using the Gronbech-Jensen-Farago time-discretization of the Langevin equation. [24; 25]. Lennard-Jones (LJ) units were used, with lengths in terms of the bead diameter \(\sigma\), and energies in terms of a LJ interaction energy scale \(\epsilon\). To connect to the parameter values of Table 1 used for sequence optimization (which are not in LJ units), we choose \(\epsilon\ =\ k_{B}T^{\star}\), where \(T^{\star}\) is a temperature on the order of the temperature below which the chains start self-assembling. For the simulations, we take parameter values from Table 1 with \(T=T^{\star}\), such that the numerical parameter values in Table 1 are also the parameter values in LJ units. With this convention, the dimensionless temperature \(\widetilde{T}\) in LJ units is \(\widetilde{T}\ =\ T/T^{\star}\). The simulation timestep was set to \(\Delta t=0.05\tau\) (where \(\tau\) is the LJ unit of time), the bead friction constant was set to \(\zeta_{0}\ =\ 100(m/\tau)\), where \(m\) and \(t\) are the LJ units of mass and time. Numbers of simulation steps are fairly arbitrary numbers, but the time in units of \(\tau\), or better yet, in units of the rotational diffusion time \(\tau_{r}\) of the rods, has physical meaning.