Deoxyribonucleic acid (DNA) is a long flexible polyelectrolyte that is housed in the aqueous environment within the cell of an organism. When a length of torsionally relaxed (untwisted) DNA is held in tension, such as is the case in many single molecule experiments, the thermal fluctuations arising from the constant bombardment of the DNA by the surrounding fluid molecules induce bending in it, while the applied tension tends to keep it extended. The combined effect of these influences is that DNA is never at its full extension but eventually attains an equilibrium value of end-to-end extension under these influences. An analytical model was developed to estimate the tension-dependent value of this extension. This model, however, does not provide any insight into the dynamics of the extensional response of DNA to applied tension nor the kinetics of DNA at equilibrium under said tension. This paper reports the results of Brownian dynamics simulations using a discrete wormlike-chain model of DNA that provide some insight into these dynamics and kinetics.