Local Cooperative Interactions Reshape the Folding Transition in a One-Dimensional Spin-Glass Model

Protein folding is the process by which a polypeptide chain organizes into its three-dimensional structure through a balance of stabilizing and destabilizing interactions encoded by the sequence. A central question in protein biophysics is how thermodynamic factors guide a polypeptide toward its native folded state despite the rugged energy landscape and the competing influence of nonnative interactions. In many biomolecular processes, cooperativity provides a mechanism by which multiple weak interactions act collectively to generate a robust response. In the context of protein folding, such cooperative effects may arise when the formation of one native contact enhances the stability or likelihood of nearby native contacts, thereby promoting collective organization toward the folded state. At the same time, folding is opposed by the much larger number of non-native interactions, whose heterogeneity can introduce frustration and destabilize folding even when the average native bias favors the folded phase. The interplay of these competing effects in determining foldability remains unclear in statistical-mechanical models. Here, we address this problem using a one-dimensional spin-glass model of protein folding with explicit shared-residue cooperative interactions encoded through wedge-based motifs. We show that modest cooperative bias can stabilize folding even where the noncooperative system remains unfolded, whereas non-native energetic fluctuation suppresses folding and shifts the transition to higher cooperative strengths. We further find that partial cooperative coverage is sufficient to lower the folding threshold. Therefore, the model provides a mean-field framework for incorporating cooperative interaction strength into the native one-dimensional model of protein folding and for describing how local cooperativity reshapes the folding transition.