Mechanics and thermodynamics of multivalent-binding induced shrinkage of hydrogels

Understanding the fundamental mechanism of shrink hydrogel sensors necessitates a complete comprehension of analyte-centered multivalent binding that occurs within their salt-rich microenvironments. However, the mechanics and thermodynamics governing this phenomenon remain insufficiently understood. Here, we aim to derive a theoretical framework that examines the impact of temporary cross-link formation on the hydrogel shrinkage due to specific binding interaction between the fixed receptors and the multivalent analytes. As a highlight of our theory, we mathematically quantify the hydrogels’ permanent and temporary cross-links using statistical thermodynamics to describe the multivalent complexation with different binding degrees while accounting for molecular-level transport factors when predicting the sensor's shrinking characteristics. Consequently, our theory unveils the upper bounds set by the external analyte concentration and analyte binding valency onto the actuation sensitivity of these sensors, whereby tuning the receptor density permits further modulation of their performances. These findings tightly correlate the microscopic properties of the analyte and hydrogel to the macroscopic behaviors of shrink sensors, facilitating a structured design regime for advanced biomedical applications.