Derivation of Multidimensional Force for Magnetic Tweezer as a Function of T₁ and T₂ Relaxation Times using Magnetic Resonance Diffusion and Langevin Equations

Abstract

Magnetic tweezer is an instrument that has the ability to exert force in terms of magnitude and direction on biomolecules, such as deoxyribonucleic acid (DNA). It is a force spectroscopy technique with wide range of applications ranging from determination of elastic properties of DNA, study of DNA-protein interactions, study of topology of DNA, study of cancer cells, among others. Magnetic field is produced when magnetic tweezer applies force on a biomolecule. The force applied is along the axis of the molecule and field direction. Several models of magnetic tweezer have been developed overtime for measurement of force applied on a particle, but each developed model is marred with limitations such as magnetic hysteresis of the core material in the tweezer and low level spatial resolution. Also, there is limited availability of knowledge on the incorporation of nuclear magnetic resonance (NMR) relaxation time constants (T₁ and T₂), with NMR diffusion and Langevin equations to determine force applied on a particle along different axis. Therefore, this study derived a multidimensional force for magnetic tweezer from T₁ and T₂ relaxation times using NMR diffusion and Langevin equations. As this model helps to overcome the limitations encountered by the previous models. The NMR diffusion equation: ∂Mᵧ/∂t = D∇²Mᵧ + (F₀/T₀)yB₁(r,t), and Langevin equation: M(dv/dt) = -ζv + X(t) were utilised in the study to derive a multidimensional force at different boundary conditions. The equations are used to derive power spectrum from T₁ and T₂ relaxation times, given that T₀ = 1/T₁ + 1/T₂. The multidimensional force derived in this study is in x, y, and z orientation with each axis having its characteristic variable. The result obtained from this study can be used to simulate T₁ and T₂ relaxation times values of DNA-binding proteins, an essential protein which enhances DNA-protein interactions. The results of the simulation can depict different features of DNA, thereby availing knowledge on DNA classification, genetic abnormality in cancer cells, among others.