Polyamines are known to induce dynamical bending of DNA molecules. This mechanism is very important since many DNA binding proteins (DNAse, transcription factor, etc.) exert their action by their ability to bend DNA. We propose an analytical model which describes the dynamical bending of DNA by polyamine ions in highly diluted DNA solutions. The bending probability depends on the entropy loss of polyamines due to their localization. This localization is facilitated by the electrostatic repulsion between multivalent counterions condensed on DNA, which reduces the entropy loss in counterion localization. Therefore DNA bending by polyamines depends on the competition between monovalent counterions and polyamines. We find that the bending probability is weak for a low binding ratio of polyamines (i.e. number of bound polyamines per base pair), whereas a high bending probability can be reached at large polyamine binding ratio. In addition, we describe a new mechanism of DNA bending. It occurs with the help of thermal agitation, which initiates the bending and favours the polyamine localization. This model provides further insights into DNA bending by polyamines and its implication in DNA condensation. A qualitative estimation of the DNA bending probability is obtained by measuring the cleavage efficiency of DNA by bleomycin versus spermidine concentration. Indeed, a local helix distortion by polyamines results in an amplification of the double-strand cleavage by bleomycin. The measurement of the bleomycin amplification is performed by analysing images of DNA molecules with atomic force microscope. Some features of the dynamical bending indicate that condensation and bending are interrelated.