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 `_. .. rubric:: 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**. .. rubric:: Design choices (at a glance) .. list-table:: :header-rows: 1 :widths: 28 72 * - **Aspect** - **Choice / Consequence** * - Sticker valency - :math:`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 :math:`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. .. rubric:: Bonds .. math:: E_{\text{bond}} = K_b \,(R - R_0)^2 with parameters: .. list-table:: :header-rows: 1 :widths: 20 40 40 * - **Symbol** - **Meaning** - **Value** * - :math:`R` - Distance between bonded beads - — * - :math:`R_0` - Equilibrium bond length - :math:`10~Å` * - :math:`K_b` - Bond spring constant - :math:`3~\mathrm{kcal\,mol^{-1}\,Å^{-2}}` .. rubric:: Angles .. math:: E_{\text{bend}} = \kappa \,\bigl(1-\cos\theta\bigr) where :math:`\theta` is the angle between three successive beads and :math:`\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. .. image:: /_static/img/specific_interaction.png :alt: Specific inter-sticker interaction energy vs distance :align: center :width: 70% .. rubric:: Switching rule * If two complementary stickers are within :math:`R_\mathrm{cut}`, they **form** a bond (probability :math:`p_\mathrm{on}=1`). * If a bonded pair reaches :math:`R \ge R_\mathrm{cut}`, the bond **breaks** (probability :math:`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. .. rubric:: Specific potential (shifted harmonic) .. math:: 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} Parameters: .. list-table:: :header-rows: 1 :widths: 22 48 30 * - **Parameter** - **Meaning** - **Value** * - :math:`E_s` - Well depth (“specific energy”); sets bond lifetime scale - user-set; reported in *kT* * - :math:`R_0` - Resting bond distance - :math:`1.122\,\sigma` * - :math:`\sigma` - Bead diameter (model length unit) - :math:`10~Å` * - :math:`R_\mathrm{cut}` - Specific bond cutoff - :math:`R_0 + 1.5~Å` * - :math:`p_\mathrm{on},\,p_\mathrm{off}` - Attempt probabilities - :math:`1,\,1` .. rubric:: Kinetics & detailed balance * With :math:`p_\mathrm{on} = p_\mathrm{off} = 1`, stochasticity stems solely from **diffusion** and the **energy landscape**; bond state is determined by :math:`R` relative to :math:`R_\mathrm{cut}`. * The **bond lifetime** scales as :math:`\tau_{\text{bond}} \propto e^{E_s/kT}`; dissociation rates show Arrhenius behavior, :math:`\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 :math:`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. .. image:: /_static/img/nonspecific_interaction.png :alt: Nonspecific LJ interaction energy vs distance :align: center :width: 70% .. math:: 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 :math:`R_\mathrm{max}` for efficiency. .. list-table:: :header-rows: 1 :widths: 22 48 30 * - **Parameter** - **Meaning** - **Value** * - :math:`E_{ns}` - LJ well depth (“nonspecific energy”); sets **contact dwell time** - user-set; reported in *kT* * - :math:`\sigma` - Bead diameter - :math:`10~Å` * - :math:`R_\mathrm{max}` - LJ cutoff - :math:`2.5\,\sigma` .. rubric:: Bonds vs. contacts — terminology * **Bonds** = **specific** sticker–sticker links (single valency, governed by :math:`E_s`). * **Contacts** = **nonspecific** LJ interactions among any beads (governed by :math:`E_{ns}`). Energy Units & Reporting ------------------------ Simulation inputs use **kcal/mol** for :math:`E_s` and :math:`E_{ns}`. For analysis and figures, energies are reported in **thermal units**: .. math:: 1~kT \approx 0.6~\mathrm{kcal\,mol^{-1}} so that :math:`E/kT` is dimensionless and temperature-explicit. Quick Reference Tables ---------------------- .. rubric:: Core parameters .. list-table:: :header-rows: 1 :widths: 22 40 38 * - **Symbol** - **Meaning** - **Default / Example** * - :math:`\sigma` - Bead diameter - :math:`10~Å` * - :math:`R_0` (bonded) - Specific bond rest distance - :math:`1.122\,\sigma` * - :math:`R_\mathrm{cut}` (bonded) - Specific bond cutoff - :math:`R_0 + 1.5~Å` * - :math:`R_\mathrm{max}` (LJ) - LJ cutoff - :math:`2.5\,\sigma` * - :math:`K_b` - Bond spring constant - :math:`3~\mathrm{kcal\,mol^{-1}\,Å^{-2}}` * - :math:`\kappa` - Bending stiffness - :math:`2~\mathrm{kcal\,mol^{-1}}` * - :math:`p_\mathrm{on},\,p_\mathrm{off}` - Specific attempt probabilities - :math:`1,\,1` * - Update cadence - Bond (create/break) evaluation interval - every **20** timesteps .. rubric:: Modeling notes * **Association** is diffusion-limited; **dissociation** requires crossing the specific energy barrier set by :math:`E_s`. * Observed dissociation decays **exponentially** with increasing :math:`E_s` (Arrhenius-like), indicating thermalization within the specific well and consistency with detailed balance.