Fig 1.
(A) Schematic depiction of an alphabet-free sticker-spacer framework for protein chains, where associative interactions, δE, between domains are sampled from an energy landscape.
The blue-red color scheme represents distinct motif IDs: blue indicates spacers, while shaded blue-red regions denote stickers with varying interaction energies. (B) Schematic representation of the probability distribution of the residues’ non-bonded interaction energy, P ( δE ) , transitioning from higher periodicity continuous sinusoidal function to systematically increasing frequencies, with randomly arranged residues shown at the bottom. (C) The Wasserstein distance of the distributions in (B) relative to the randomly distributed residues decreases as frequency increases.
Fig 2.
Schematic description and simulation snapshots of the sequence-specific spatial arrangement of residues.
The variations of i) periodically distributed sticker-spacer motifs ,
, and
, and ii) randomly distributed sticker-spacer motifs along the chain backbone
,
, and
are shown in (A). Conformations of the FELaS chain in very dilute solutions are depicted in (B). The interaction strengths between distinct stickers are illustrated in the pairwise interaction matrix in (C), it depicts the red motif exhibiting an interaction energy of ε = 1, representing the highest sticker strengths. In contrast, a blue motif denotes spacers with the lowest interaction energy. (D) As the periodicity of the sticky region along the chain increases, intermediate sticker strengths are introduced, conserving total energy along the chain. Simulation snapshots of the bulk system of chains with periodic and randomly distributed stickers while fixing all other parameters. A membrane-shaped layer forms for
, a cluster domain for
, and a larger sticky cluster for
. The structural architecture transitions into a homogeneous condensed phase for randomly distributed
,
, and
.
Fig 3.
Equilibrium and non-equilibrium properties of the programmable energy landscape sticker-spacer model: A) The phase diagram of periodically distributed chains (blue shaded) condensates reveals that exhibits a critical temperature
more than two-fold higher than that of
due to the lower periodic occurrence of the sticker.
Conversely, the randomly distributed (red shaded) for the same systems shows a lower critical temperature . B) Anomalous peaks in the structure factor S(q) of periodically distributed chains
characterize equilibrium folded sticky domain sizes absent in randomly distributed chains. C) With increasing period, periodically distributed chains
displays more glassy arrested behavior, following
, whereas randomly distributed chains follows
. D) The complex modulus G of randomly distributed chains decays faster than periodically distributed chains. E) Elastic (
) and viscous (
) moduli exhibit a high-frequency viscous-dominated regime and an elastic-dominated crossover at an intermediate frequency range. For ω → 0, a Maxwell viscous-dominated trend is observed for randomly distributed chains, while a Kelvin-Voigt
plateau is observed for periodically distributed chains. F) periodically distributed systems are highly viscous, with viscosity increasing with the periodicity of the stickers, whereas randomly distributed systems have an order of magnitude lower viscosity.
Fig 4.
Simulation snapshots depict the transition from ordered to disordered energy landscapes of monomer interactions for various sticker and spacer arrangements. Alongside periodically distributed (A & B) and randomly distributed
(A & B) chains, two new sticker-spacer arrangements are introduced.
(A): sticker residues are shuffled while keeping spacer periodicity fixed, and its corresponding bulk simulation snapshot is shown in (B).
(A & B): an alternative sticker-spacer chain arrangement and its corresponding bulk simulation snapshot shown here. (C) shows the phase diagram of the four different sequences pattering where
of
is almost double of
. (D) Static structure factors S(q) for condensates formed by different sticker-spacer arrangements. (E) Mean square displacement averaged over all beads and shown for condensates formed by different sticker-spacer arrangements. (F) Elastic and viscous moduli for different sticker-spacer arrangements. (G) The viscosity was computed for the different sticker-spacer arrangements.
Fig 5.
Comparison of the experimental results of reported in [21] and [40] in (A) with our model of periodic and random sequences in (B) as a function of periodicity of the sequence.
Both experimental and simulation results show an increase in viscoelasticity as the periodicity of the arranged sequences increases.
Fig 6.
Tangent-tangent correlation in condensates was calculated for the periodically distributed (A–E) and randomly distributed (F) stickers.
(A) For , periodic sticky regions aggregate; however, the chains show uniform exponential decay correlation. (B)
shows anomalous exponential decay in
, indicating a folded structure. (C)
displays diverse position-dependent residue characteristics and long-range order; spacer residues show exponential decay, while spool-like folded domains show non-exponential decay and negative correlation. (D)
also shows anomalous exponential decay, and (E)
has distinct exponential decay lengths. (F) A collapsed exponential decay in the tangent correlation of randomly distributed chains (
,
,
) exhibits worm-like chain behavior with a persistence length of
.
Fig 7.
The self part of the van Hove distribution function of the sticker residues, , are shown with the color scheme indicating the temporal evolution: dark colors represent the earlier distribution (t ~ 10τ), whereas light colors depict the late-time behavior (
).
All the curves correspond to distributions taken at the same time intervals. Gaussian fits are shown for the early time (t ~ 10τ, blue dashed line) and late time (, red solid line). (A–C) represent chains with periodic sticker patterns having periods 10, 25, and 100, which remain Gaussian at long times. (D) shows
chains with a sticker periodicity of 25 and shuffled stickers, which also remain Gaussian at long time. However, (E) depicts
chains with a sticker periodicity of 25 and alternating sticker-spacer arrangements, which deviate from Gaussian behavior. (F) corresponds to R25 (other
shows similar behavior), chains with randomly arranged beads, which fail to exhibit Gaussian behavior at long times, indicating the lack of spatial organization in the system.
Fig 8.
Biomolecular condensate phases of at different number densities ρ (by varying the box volume
) creates a range of architectures, from a network fluid structure to clusters of micelles.
(G) Orthogonal radius of gyration of sticky clusters , (H) average number of motifs in a single cluster,
, and (I) the average number of sticky clusters in bulk,
, as a function of box dimensions ρ are shown here. The vertical lines separate the percolated single and multiple clusters in the bulk phase.
Fig 9.
The first row displays snapshots of sticker clusters for four different chain lengths N: (A) 25, (B) 50, (C) 100, and (D) 200, with individual clusters depicted in various colors.
In the second row, corresponding systems for each chain length illustrate how cluster shapes evolve across different density regimes, transitioning from spherical to elongated micelles. The third row presents the probability distribution of the radius of gyration, , for the sticker clusters.