From Heteropolymer Stiffness Distributions to Effective Homopolymers: A Conformational Analysis of Intrinsically Disordered Proteins

TL;DR

This work addresses how spatial heterogeneity in bending stiffness along polymers affects large-scale conformations and whether the chain can be described by a single effective persistence length $\ell_{p,\text{eff}}$. It develops a theoretical framework that maps a heteropolymer with Gaussian-distributed bending parameters $\kappa_i$ to a homogeneous chain, yielding $\kappa_{\text{eff}}=\kappa_0-\frac{\sigma_\kappa^2}{2}h(\kappa_0)$ and $\ell_{p,\text{eff}}<\langle \ell_p\rangle_{\text{dis}}$, with a leading $\sigma_p^2$-dependent correction; correlations are captured by $c_{\text{eff}}$. The authors validate the theory with extensive off-lattice Monte Carlo simulations for both ideal and self-avoiding chains, finding excellent agreement for narrow distributions ($\sigma_\kappa/\kappa_0 \lesssim 0.1$) and moderate $\ell_p$; deviations grow modestly for broader distributions. They also explore coarse-grained models of intrinsically disordered proteins (IDPs), showing IDP shapes align with mapped homopolymers but are slightly larger in ideal models and more compact when excluded volume is included, highlighting the role of intramolecular interactions not captured by the base theory. The results provide a practical route to simplify heterogeneous biopolymers and set the stage for applications to IDPs, with open data and code available.

Abstract

Synthetic copolymers and biopolymers, such as polypeptides and double-stranded DNA, often exhibit strong variations in bending stiffness along their contour, which can significantly impact conformational behavior at larger scales. To investigate these effects, we employ a discretized heterogeneous worm-like chain model, where the local persistence lengths are drawn from a Gaussian distribution. In the first part, we develop a theoretical model that maps such heterogeneous chains to homogeneous chains with a single effective persistence length. For uncorrelated disorder, our model predicts that this effective stiffness is systematically smaller than the arithmetic mean of the local persistence lengths, indicating that flexible segments have a bigger influence on the overall chain stiffness than rigid segments. We validate our model predictions using off-lattice Monte Carlo simulations, considering both ideal and self-avoiding chains in good solvent, and find excellent agreement in the regime, where the persistence lengths are on the order of a few bond lengths, consistent with typical values observed in polypeptides. In the second part, we performed simulations using various coarse-grained models of intrinsically disordered proteins (IDPs), finding that the simulated IDPs have similar shapes like the corresponding homogeneous and heterogeneous worm-like chains. However, the IDPs are systematically larger than ideal worm-like chains, yet slightly more compact when excluded volume interactions are considered. We attribute these differences to intramolecular interactions between non-bonded monomers, which our theoretical models do not account for.

Figures (6)

• Figure 1: Schematic representation of a bead-spring polymer model, consisting of $N=20$ spherical monomers of diameter $d$, connected by harmonic springs of rest length $b$. The angle $\theta$ indicates the bending angle between three consecutive beads along the chain, while the segment color indicates the local bending stiffness.
• Figure 2: (a) Probability distributions of the bending potential parameter, $P(\kappa)$, for selected mean $\kappa_0$ and standard deviations $\sigma_\kappa$. (b) Corresponding distributions of the persistence length, $P(\ell_\text{p})$.
• Figure 3: Predicted effective persistence length $\ell_\textrm{p,eff}$ according to Eq. \ref{['eq:lpeffpre']} normalized by the average persistence length $\left\langle \ell_\textrm{p} \right\rangle_{\text{dis}}$ versus reduced standard deviation $\sigma_\text{p}/\left\langle \ell_\text{p} \right\rangle_{\text{dis}}$ for varying means $\kappa_0$ (color coding). The inset shows the difference between predictions of $\ell_\text{p,eff}$ from the full theory, Eq. \ref{['eq:lpeffpre']}, and from the linearized theory, Eq. \ref{['eq:lpeff']}, normalized by $\left\langle \ell_\textrm{p} \right\rangle_{\text{dis}}$.
• Figure 4: (a) Statistically averaged end-to-end distance $R_\text{ee}$ (left $y$-axis) and radius of gyration $R_\text{g}$ (right $y$-axis), as well as (b) asphericity $\text{AS}$, acylindricity $\text{AC}$, and shape anisotropy $\text{A}$ for homogeneous worm-like chains with $N=100$ beads as a function of persistence length $\ell_\text{p}$. The solid and dashed lines show results for chains without and with excluded volume interactions, respectively.
• Figure 5: Relative difference $\Delta O$ between the disorder-averaged configurational properties of the heteropolymers and the ensemble-averaged properties of homopolymers with effective persistence length $\ell_\text{p,eff}$. Panels (a) and (b) show data for chains without and with excluded volume interactions, respectively. The angle potential parameters $\kappa_i$ of the heteropolymers are Gaussian distributed with mean $\kappa_0$ and variance $\sigma_\kappa^2$ (see main text), resulting in $\ell_\textrm{p,0}=\ell_\textrm{p}(\kappa_0)$. The symbols are colored according to the values of $\sigma_\kappa/\kappa_{0}$ as indicated. Error bars indicate the standard error of the mean, determined from 100 different realizations of the disorder.