Model Overview

This page describes a typical set of model parameters (force-fields) used in the sticker spacer simulations. The values are mostly obtained from Chattaraj 2024, Chattaraj 2025.

Scope & Rationale

We typically model heterotypic interactions between complementary sticker types (e.g., inspired by SH3–PRM or SUMO–SIM systems), where intersticker crosslinks generate an intra-condensate network.
A two-component (A & B) system serves as a minimal model for multicomponent biological condensates.
Insights are expected to transfer to homotypic (single-component) condensates provided the biopolymer follows a sticker–spacer architecture.

Design choices (at a glance)

Aspect	Choice / Consequence
Sticker valency	\(1\) (saturating); a sticker can participate in only one specific bond at a time.
Specific vs nonspecific	Specific = reversible bonds between complementary stickers; nonspecific = Lennard–Jones contacts between any beads.
Update cadence	Bond creation/breaking rules evaluated every 20 timesteps to allow local relaxation after formation.
Energies	Inputs in kcal/mol; reported in kT with \(1~kT \approx 0.6~\mathrm{kcal\,mol^{-1}}\).

Polymer Force-Fields

Connectivity and flexibility within each chain are enforced with harmonic bonds and cosine bending terms.

Bonds

\[E_{\text{bond}} = K_b \,(R - R_0)^2\]

with parameters:

Symbol	Meaning	Value
\(R\)	Distance between bonded beads	—
\(R_0\)	Equilibrium bond length	\(10~Å\)
\(K_b\)	Bond spring constant	\(3~\mathrm{kcal\,mol^{-1}\,Å^{-2}}\)

Angles

\[E_{\text{bend}} = \kappa \,\bigl(1-\cos\theta\bigr)\]

where \(\theta\) is the angle between three successive beads and \(\kappa = 2~\mathrm{kcal\,mol^{-1}}\) controls bending stiffness.

Specific (Sticker–Sticker) Interactions

Complementary stickers interact via reversible, saturating bonds (i.e., valency = 1). Bond formation/breaking depends only on inter-sticker distance under the settings below.

Specific inter-sticker interaction energy vs distance

Switching rule

If two complementary stickers are within \(R_\mathrm{cut}\), they form a bond (probability \(p_\mathrm{on}=1\)).
If a bonded pair reaches \(R \ge R_\mathrm{cut}\), the bond breaks (probability \(p_\mathrm{off}=1\)).
While bonded, the nonspecific LJ between the pair is disabled and replaced by the specific potential. Upon bond break, the LJ potential is reinstated.

Specific potential (shifted harmonic)

\[\begin{split}E_{\text{spec}}(R) = \frac{E_s}{(R_0 - R_\mathrm{cut})^2} \left[(R - R_0)^2 - \bigl(R_\mathrm{cut} - R_0\bigr)^2\right], \quad \begin{cases} E_{\text{spec}}(R_0) = -E_s,\\[2pt] E_{\text{spec}}(R_\mathrm{cut}) = 0,\\[2pt] E_{\text{spec}}(R>R_\mathrm{cut}) = 0~. \end{cases}\end{split}\]

Parameters:

Parameter	Meaning	Value
\(E_s\)	Well depth (“specific energy”); sets bond lifetime scale	user-set; reported in kT
\(R_0\)	Resting bond distance	\(1.122\,\sigma\)
\(\sigma\)	Bead diameter (model length unit)	\(10~Å\)
\(R_\mathrm{cut}\)	Specific bond cutoff	\(R_0 + 1.5~Å\)
\(p_\mathrm{on},\,p_\mathrm{off}\)	Attempt probabilities	\(1,\,1\)

Kinetics & detailed balance

With \(p_\mathrm{on} = p_\mathrm{off} = 1\), stochasticity stems solely from diffusion and the energy landscape; bond state is determined by \(R\) relative to \(R_\mathrm{cut}\).
The bond lifetime scales as \(\tau_{\text{bond}} \propto e^{E_s/kT}\); dissociation rates show Arrhenius behavior, \(\text{Rate}\propto e^{-E_s/kT}\) (consistent with thermal equilibration inside the well).
Bond creation/breaking rules are evaluated once every 20 timesteps to allow newly formed pairs to relax near \(R_0\).

Nonspecific (All-Bead) Interactions

All bead pairs (stickers and spacers) experience an isotropic Lennard–Jones (LJ) interaction that enforces excluded volume and a moderate attraction.

Nonspecific LJ interaction energy vs distance

\[E_{\text{LJ}}(r) = 4\,E_{ns} \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]\]

with a truncation at \(R_\mathrm{max}\) for efficiency.

Parameter	Meaning	Value
\(E_{ns}\)	LJ well depth (“nonspecific energy”); sets contact dwell time	user-set; reported in kT
\(\sigma\)	Bead diameter	\(10~Å\)
\(R_\mathrm{max}\)	LJ cutoff	\(2.5\,\sigma\)

Bonds vs. contacts — terminology

Bonds = specific sticker–sticker links (single valency, governed by \(E_s\)).
Contacts = nonspecific LJ interactions among any beads (governed by \(E_{ns}\)).

Energy Units & Reporting

Simulation inputs use kcal/mol for \(E_s\) and \(E_{ns}\). For analysis and figures, energies are reported in thermal units:

\[1~kT \approx 0.6~\mathrm{kcal\,mol^{-1}}\]

so that \(E/kT\) is dimensionless and temperature-explicit.

Quick Reference Tables

Core parameters

Symbol	Meaning	Default / Example
\(\sigma\)	Bead diameter	\(10~Å\)
\(R_0\) (bonded)	Specific bond rest distance	\(1.122\,\sigma\)
\(R_\mathrm{cut}\) (bonded)	Specific bond cutoff	\(R_0 + 1.5~Å\)
\(R_\mathrm{max}\) (LJ)	LJ cutoff	\(2.5\,\sigma\)
\(K_b\)	Bond spring constant	\(3~\mathrm{kcal\,mol^{-1}\,Å^{-2}}\)
\(\kappa\)	Bending stiffness	\(2~\mathrm{kcal\,mol^{-1}}\)
\(p_\mathrm{on},\,p_\mathrm{off}\)	Specific attempt probabilities	\(1,\,1\)
Update cadence	Bond (create/break) evaluation interval	every 20 timesteps

Modeling notes

Association is diffusion-limited; dissociation requires crossing the specific energy barrier set by \(E_s\).
Observed dissociation decays exponentially with increasing \(E_s\) (Arrhenius-like), indicating thermalization within the specific well and consistency with detailed balance.