Fractional kinetic modelling of the adsorption and desorption process from experimental SPR curves

TL;DR

This work shows that conventional integer-order kinetics inadequately describe SPR-based adsorption-desorption, motivating a Caputo fractional-order kinetic model that accounts for non-local memory and transport effects. Analytic solutions in terms of Mittag-Leffler functions are derived for both association and dissociation phases, with extensions to variable-order forms. Applied to IBP–Congo Red SPR data across $[A]=7.5$–$97.5\,\mu$M at $pH=7.4$ and $16^\circ C$, the fractional model, especially Case 3 with $\alpha\approx 0.6$ and recalibrated $k_a$, $k_d$, yields a substantial reduction in the objective cost ($E$) compared to the integer-order fit. The results underscore memory effects and non-local dynamics in SPR kinetics and suggest that fractional-order formulations can markedly improve interpretation and parameter estimation in surface-binding experiments.

Abstract

The application of surface plasmon resonance (SPR) has transformed the field of study of interactions between a ligand immobilized on the surface of a sensor chip, designated as $L_S$, and an analyte in solution, referred to as $A$. This technique enables the real-time measurement of interactions with high sensitivity. The dynamics of adsorption-desorption process, $A+L_S \rightarrow AL_S$, can be expressed mathematically as a set of coupled integer-order differential equations. However, this approach has limited ability to acoount for temperature distribution, diffusion and transport effects involved in the reaction process. The fractional kinetic model provides a methodology for incorporating non-local effects into the problem. In this study, the proposed model was applied to analyze data to the interaction between Immobilized Baru Protein (IBP) and Congo Red dye (CR) at concentrations ranging from $7.5$ to $97.5$ $μM$, at pH $7.4$ and $16^o$ C. The variation in the kinetic constants was studied, and it was demonstrated that the integer-order model is unable to adequately represent the experimental data. This work has shown that the fractional-order model is capable of capturing the complexity of the adsorption-desorption process involved in the SPR data.

Figures (7)

• Figure 1: Experimental data for an analyte concentration of $75\mu M$ is represented by star symbol. The solid line represents the fit using the strategy labeled Case 1, while the dashed line shows the result obtained from the path labeled Case 2.
• Figure 2: Study of the variation of $k_a$. The solid line represents the result obtained by fitting the integer-order model. The thick-dashed line represents the result for a $+20\%$ variation in $k_a$ (with $k_d$ held constant). Similarly, the thin-dashed line represents a $-20\%$ variation in $k_a$. The thick-dotted line represents a $+40\%$ variation in $k_a$, while the thin-dotted line represents a $-40\%$ variation in $k_a$.
• Figure 3: The solid line represents the result obtained by fitting the integer-order model. The thick-dashed line represents the result for a $+20\%$ variation in $k_d$, i.e., $k_d=k_d(1+0.2)$ (with $k_a$ held constant). Similarly, the thin-dashed line represents a $-20\%$ variation in $k_d$. The thick-dotted line represents a $+40\%$ variation in $k_d$, while the thin-dotted line represents a $-40\%$ variation in $k_d$.
• Figure 4: Study of the variation of $\alpha$ (fractional-order), while keeping $k_a$ and $k_d$ fixed. The solid line represents the result obtained by fitting the integer-order model. The thick-dashed line corresponds to $\alpha=0.9$ (with $k_a$ and $k_d$ held constant). Similarly, the thin-dashed line represents $\alpha=0.8$, the thick-dotted line corresponds to $\alpha=0.7$, and the thin-dotted line represents $\alpha=0.6$.
• Figure 5: Multivariable adjustment of $k_a$, $k_d$, and $\alpha$. Dotted line represents $\alpha=1$ with $E=42.3$. The solid line corresponds to $\alpha=0.58$, with $E=1.47$. The star symbols denote the experimental data.