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Strongly correlated electrostatics of DNA systems has drawn the interest ofmany groups, especially the condensation and overcharging of DNA by multivalentcounterions. By adding counterions of different valencies and shapes, one canenhance or reduce DNA overcharging. In this papers, we focus on the effect ofmultivalent co-ions, specifically divalent co-ions such as SO$_4^{2-}$. Acomputational experiment of DNA condensation using Monte$-$Carlo simulation ingrand canonical ensemble is carried out where DNA system is in equilibrium witha bulk solution containing a mixture of salt of different valency of co-ions.Compared to system with purely monovalent co-ions, the influence of divalentco-ions shows up in multiple aspects. Divalent co-ions lead to an increase ofmonovalent salt in the DNA condensate. Because monovalent salts mostlyparticipate in linear screening of electrostatic interactions in the system,more monovalent salt molecules enter the condensate leads to screening out ofshort-range DNA$-$DNA like charge attraction and weaker DNA condensation freeenergy. The overcharging of DNA by multivalent counterions is also reduced inthe presence of divalent co$-$ions. Strong repulsions between DNA and divalentco-ions and among divalent co-ions themselves leads to a {\em depletion} ofnegative ions near DNA surface as compared to the case without divalentco-ions. At large distance, the DNA$-$DNA repulsive interaction is stronger inthe presence of divalent co$-$ions, suggesting that divalent co$-$ions role isnot only that of simple stronger linear screening.

DNA like$-$charge attraction and overcharging by divalent counterions in the presence of divalent co$-$ions