Feynman-Kac equation for microtubule search process in prometaphase

In prometaphase, microtubules form the spindle structure through dynamic instability, accurately locating and capturing chromosomes to ensure the equal distribution of genetic material (DNA on chromosomes). In this paper, we establish a dynamic model of the microtubule growth-rotation search process by considering the non-local polymerization and depolymerization of microtubules affected by mechanisms such as katanin proteins and kinetochore fibers and derive the macroscopic equation for the microtubule-kinetochore capture model by imposing appropriate boundary conditions. Furthermore, we derive the Feynman-Kac equations that govern the probability density of the functional for the microtubule growth-rotation search model. Finally, we validate the derived equations by comparing results from using a deep learning method to solve the equations with those from applying Monte Carlo simulations to the microscopic models.