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Analyzing the kinetics of biological processes plays a significant role in understanding fundamental cellular functions. Many physics-based technologies used to study such processes are limited by the shot noise inherent to the coherent states of light. These technologies can greatly benefit from leveraging quantum probes to improve the sensitivity of measurements in cellular biology. The surface plasmon resonance technique has been used effectively to achieve label-free, real-time measurements of protein binding kinetics, which constitutes an important biological phenomenon occurring near the cell membrane. Here, we demonstrate the integration of this technique with the two-mode bright squeezed state having fewer fluctuations as compared to the coherent state to improve the sensitivity of measurement in studying a protein-gold adsorption process. We show 4 dB of squeezing as we record the signal-to-noise ratio as the function of time, and it is maintained throughout the kinetic process. The improved signal-to-noise ratio leads to a $60\%$improvement in the sensitivity of measuring the observable rate constant $k_s$. The quantum advantage as shown in terms of squeezing is achieved despite the total absorption of $74\%$from the source until the final detection after the sensor. Overall, we provide the most practical setup for improving the sensitivity of the time-dependent measurements involved in various biological processes at the molecular level.