Temperature Effects on Threshold Counterion Concentration to Induce Aggregation of fd Virus

We seek to elucidate the dominant mechanism of attractive interaction between like-charged biopolymers by measuring the temperature dependence of the critical divalent counterion concentration ($C_c$) for the aggregation of fd viruses. A decrease in either temperature or the dieletric constant alone causes a decrease in $C_c$, providing evidence for the Wigner crystal model. Surprisingly, the effects of these two parameters can be combined so that $C_c$ is expressed as a function of a single parameter: the Bjerrum length. $C_c$ decreases exponentially as the Bjerrum length increases, suggesting that an energetic balance between the entropic effect of counterions and the counterion mediated attractive interaction gives rise to the onset of bundle formation.

Figure 1

An increase in temperature ($T$) leads to a decrease in dielectric constant ($\epsilon$) of the solvent, thus causing $C_c$ to decrease. (a) When the $\epsilon$ of the solvent is not a fixed value, the $C_c$ for either ${\mathrm{MgCl}}_2$ (●) or ${\mathrm{CaCl}}_2$ (▴) decreases with increasing $T$. As $T$ increases, $\epsilon$ (■) drops. (b) $C_c$ for ${\mathrm{MgCl}}_2$ (●) decreases with the amount of ethanol at 20° C. Addition of ethanol leads to a decrease in $\epsilon$ (■).

Figure 2

Effect of temperature ($T$) on $C_c$ under fixed dielectric constants ($\epsilon$) and fixed $l_B$. (a) At $\epsilon =73.2$ (●) and $\epsilon =69.9$ (■), $C_c$ increases with increasing $T$. (b) When the solution $\epsilon$ is adjusted to fix $l_B$, $C_c$ does not vary with $T$. At fixed Bjerrum lengths, $C_c$ is 45 mM with $l_B=7.28$ (●), and 40 mM with $l_B=7.38$ (■).

Figure 3

$C_c$ as a function of $l_B$. Results for ${\mathrm{MgCl}}_2$ (▴) and ${\mathrm{CaCl}}_2$ (●) indicate that $C_c$ decreases exponentially with increasing $l_B$. Dashed lines are fits to the experimental data with $\ln C_c=-A{l}_B+B$.