Sequence Blockiness Controls the Structure of Polyampholyte Necklaces

A scaling theory of statistical (Markov) polyampholytes is developed to understand how sequence correlations, that is, the blockiness of positive and negative charges, influences conformational behavior. An increase in the charge patchiness leads to stronger correlation attractions between oppositely charged monomers, but simultaneously, it creates a higher charge imbalance in the polyampholyte. A competition between effective short-range attractions and long-range Coulomb repulsions induces globular, pearl-necklace, or fully stretched chain conformations, depending on the average length of the block of like charges. The necklace structure and the underlying distribution of the net charge are also controlled by the sequence. Sufficiently long blocks allow for charge migration from globular beads (pearls) to strings, thereby providing a nonmonotonic change in the number of necklace beads as the blockiness increases. The sequence-dependent structure of polyampholyte necklaces is confirmed by molecular dynamics simulations. The findings presented here provide a framework for understanding the sequence-encoded conformations of synthetic polyampholytes and intrinsically disordered proteins (IDPs).


Boundary III/IV between the necklaces with the net charge in beads and strings: Comparison of the free energies
The described structure of both types of necklaces, with charge in beads III and charge in strings IV, allows for deriving the boundary III/IV between them Λ b/s f 1/9 N 1/3 u 4/9 (S1) using the free energy comparison. If necklace IV with the charged strings also forms below the boundary, at Λ < Λ b/s , it would have a higher number of beads as compared to necklace III with the charged beads. This results in the excess surface energy per bead and N bead F surf bead ∼ Λ −1/6 per PA chain. Here we used D bead and N bead found for necklace IV and given by eqs. 27 and 25. The energy gain in necklace IV is provided by negligible Coulomb self-energy of beads, which for necklace III is given by eq. 16 and equals per bead and N bead F bead−self Coul ∼ Λ 1/3 per PA. Thus, at Λ ≤ Λ b/s , the transition between necklaces III and IV is accompanied by the Coulomb energy gain ∼ Λ 1/3 but the surface energy cost ∼ Λ −1/6 and therefore takes place at increasing Λ. Owing to the continuously changing N bead , the balance between the energy gain and lost reads F surf bead F bead−self Coul , with the energies per per bead given by eqs. S2 and S3. This leads to the boundary position defined by eq. S1 and eq. 23 of the main text.

The details of molecular dynamics simulations
Simulations were performed using LAMMPS package 1 and following our earlier work 2. In brief, Markov PAs were modelled as the particles connected by finitely extensible nonlinear elastic (FENE) springs and interacting via truncated and shifted Lennard-Jones potential 6 . The values of the parameters ε LJ = 0.34k B T , R c = 2.5σ, K = 7k B T /σ 2 , and R 0 = 2σ were chosen to provide Θ solvent conditions. 3,4 Ionic monomers interact with each other via a bare Coulomb potential with z i,j = ±1 being their charge valence and r ij being the distance between these monomers.
Bjerrum length is set to l b /σ = 3, and solvent is considered implicitly, as the continuous dielectric medium.
Sequences of PAs with f = 1 were generated from the Markov process with the correlation parameter values, λ = −0.5, 0, and 0.5. For each λ, these sequences provide the ensemble of statistically neutral PAs, i.e., the ensemble-averaged charge of these PAs is zero. Then, the sequences (Markov process realizations) providing the required nonzero global charge of the PA, Q, were chosen to perform further simulations. To provide reliable averaging over the sequence realizations, simulations were performed for N PAs with different sequences. N = 10 was sufficient to obtain the average PA gyration radius independent of the given sequence realizations.

S3
The time step for simulations was equal to 0.005τ LJ , and the temperature of the system, T = 1, was maintained by the Langevin thermostat, and the damping factor was equal to damp = 100τ LJ . 2 For each PA realization, the MD run consisted of 5×10 6 equilibration steps (2.5 × 10 4 τ LJ ) followed by 1.5 × 10 7 productions steps (7.5 × 10 4 τ LJ ) to get the PA gyration radius, R g . The equilibrium sampling is ensured by the analysis of the autocorrelation function of R g . 2,4 For N = 256, the relaxation times were in the range of (1 − 3) × 10 4 steps, i.e., (50 − 150)τ LJ , depending on the sequence and charge blockiness, λ. For N = 1024, the relaxation times were estimated to be (3−9)×10 4 steps, i.e., (150−450)τ LJ . Higher relaxation times were observed for higher λ values. We emphasize that the relaxation time is strongly dependent on the sequence realization, and our estimates correspond to the characteristic ensemble-averaged values. The necklace structures were found to be strongly fluctuating.
During the simulation run, PAs switched between conformations with the different number of beads many times, thereby confirming the reliability of our averaging. 5,6 For N = 256 and N = 1024, the global charge value was chosen to be Q = 20 and Q = 40, respectively. These charge values are on the order of the characteristic global charge (absolute value) of Makrov statistically neutral PAs, √ f N Λ, and therefore do not strongly bias their charge statistics. 7 The resulting values of R g are summarized in Table S1 and conclusively demonstrate that PA conformations are strongly sequence-dependent. For both short and long PAs, their dimensions decrease at increasing charge blockiness, λ.