According to the Poisson– Boltzmann (PB) theory, a charge is treated as a point charge and only the Coulomb interactions are considered, whereas the correlation between the charges is ignored. The charges on the surface of the polyelectrolyte are deemed to be evenly distributed, i.e. a mean field. The surface charge distribution is determined by the competition between electrostatic interactions and ion entropy in bulk solution. The Poisson equation is used to define the electrostatic interactions and the Boltzmann distribution determines the counterion distribution. The boundary condition is that the entire system is neutral. Polyelectrolytes such as DNA are often approximated as an isolated, infinite long cylinder. The cylindrical PB equation can then be used to obtain the counterion distribution. However, since this theory ignores many factors such as the excluded volume and the interactions between the counterions (correlation effect), it fails to explain the attraction between surfaces with like charges as is the case in DNA condensation.

In solution, DNA molecules coexist with counterions. However, how the counterions distribute around DNA and the conditions for counterion condensation on the DNA surface is worth considering. It should be noted that the concept of “ counterion condensation” used by Manning is completely different from “ DNA condensation” . The former refers to the aggregation of counterions on the DNA surface due to its electric field thus forming an electric double-layer-like structure. Manning defined counterion condensation as non-localized binding (or territorial binding). This type of binding is characterized by the ions within the bound region retaining mobility and the existence of a diffuse chemical potential stemming from the concentration difference with the bulk solution. In contrast, the concept of localized binding is characterized by the ions binding closely to the polyelectrolyte resulting in a lack of mobility. The Manning theory was developed during the 1960s to explain counterion condensation on polyelectrolytes based on the concept of non-localized binding.^[42] The theory is concise and has been verified by many experiments. Here we detail its main points.

Consider a polyelectrolyte chain consists of Z_p-valent charged groups. Every charged group is seen as a point charge carrying Z_pe with a mean separation b. Suppose the valence of the counterion is Z_c. Define β = Z_pe/b, ξ = e²/ε k_BTb = l_B/b, where l_B is the Bjerrum length. The latter represents the Manning parameter, which is an estimation of the charge density of a polyelectrolyte (note that a Gaussian unit is used). Approximating the polyelectrolyte as an infinite long line charge, the electrostatic energy of a counterion which is r distant from the line charge is given by u_ip = − 2Z_ceβ ln (r)/ε . The contribution A_i (r₀) to the phase integral of the region in which the counterions are at a distance r ≤ r₀ is

where f(r₀) is a finite factor. If ξ ≥ | Z_cZ_p| ^{− 1}, equation (1) diverges. To avoid this, counterions tend to condense on a polyelectrolyte surface to screen its charges. Then, the effective charge q_net is lowered from q = Z_pe to q_net = (1 − | Z_c| θ _c)Z_pe = (1 − | Z_c| θ _c)q, where θ _c is the number of counterions bound to every charged group. Consequently, the mean charge separation is increased, which results in ξ _net = (1 − | Z_c| θ _c)ξ . When ξ _net is lowered to a critical value | Z_cZ_p| ^{− 1}, the counterion condensation reaches equilibrium. For DNA, Z_p = − 1, b = 1.7 Å , T is set at 298 K (room temperature), ε is set as the dielectric constant of water 82, and ξ is calculated to be 4.2, which is greater than | Z_cZ_p| ^{− 1}. Based on this conclusion we get:

Alternatively, define the charge fraction f = 1 − | Z_c| θ _c, which identifies the fraction of charges in the polyelectrolyte which are not neutralized. Hence

for monovalent counterions such as Na⁺ , f = 0.24; for divalent counterions such as Mg²⁺ , f = 0.12.

In summary, the main conclusion of the Manning condensation theory is that if the charge density of a polyelectrolyte is greater than a critical value, the counterions will condense on the polyelectrolyte surface to lower the charge density to a given value. This type of counterion condensation occurs when the Coulomb electrostatic potential exceeds the influence of counterion entropy.

If the bulk solution contains two types of counterions, such as monovalent and divalent counterions, Manning’ s deduction provides the following equation for describing θ ₁ and θ ₂:

In the above equations, Z₁, c₁, and θ ₁ represent the valence, mole concentration, the volume of the binding region of one mole polyelectrolyte, and the number of counterions bound to every charged group referring to monovalent ions, respectively. Z₂, c₂, and θ ₂ are the corresponding values referring to divalent ions. V_p(z) is the volume of the Z-valent counterion binding region for every mole of polyelectrolyte with a unit of cm³/mol. κ is the inverse of the Debye– Hü ckel screening parameter. Equations (4) and (5) can be solved by numerical iteration, hence the charge fraction f = Z₁θ ₁ + Z₂θ ₂.

The Manning theory applies excellently to low-valent counterion (such as Na⁺ , Mg²⁺ ) cases. Various titration experiments have shown that the neutralized charge fraction in DNA agrees well with the theory. However, when the counterion valence is three or higher, the theory can no longer explain the induced DNA condensation phenomenon. In DNA condensates, the fraction of neutralization no longer maintains specific values such as f = 1 − (3ξ )^{− 1} ≅ 92% (for trivalent ions). On the contrary, the fraction may increase to such a level that the effective charge of the DNA is reversed, i.e. charge inversion. Moreover, similar to the PB theory, the Manning theory is unable to explain the phenomenon of like-charge DNA surfaces being attracted to each other during the condensation process. Despite this, the Manning theory is still a successful theory proposed on the basis of Occam’ s razor; it provides concise testable conclusions based on several premises. Numerous experiments have verified the theory; Section 8 illustrates that the theory can even be applied to interpret reentrant condensation behavior during multivalent ion-induced DNA condensation. In cases of a non-applicable phenomenon, some premises may no longer be sustained; for example, the binding is no longer non-localized.

A feasible explanation as to why the Manning theory fails to explain cases of multivalent ions is based on the counterion correlation theory.^{[7, 56]} The premise of the theory is that the electrostatic correlation between the counterions is significant when the valence of the counterion is three or higher, in comparison to monovalent and divalent counterions, where thermal fluctuation counterbalances the electrostatic correlation. Coulomb’ s coupling parameter is defined as:

where Z is the valence of the counterion, k_BT is thermal energy, R is the mean distance between the counterions. For a Wigner crystal (Fig. 4), π R² = 1/n (n = σ /Ze, is the counterion concentration), . For monovalent counterions, Γ ∼ 1, the correlation is weak and the mean field theory is valid. However, for greater Z-valent ions, the mean field theory can no longer be applied. Specifically, for Z = 3, DNA charge density σ = 1.0 e/nm², Γ = 6.4, signifying a strong correlation effect. At this point, electric fields induced by a bound counterion tend to resist approaching counterions; i.e. electrostatic or counterion correlation is involved.

Fig. 4.

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	Fig. 4. The Wigner– Seitz crystal.[7]

Grosberg, Nguyen, and Shklovskii introduced the Wigner Crystal concept to explain electrostatic correlation. The main result of the Wigner theory is that a gas of electrons moving in a uniform, inert, and neutralizing background (the plasma system) will crystallize and form a lattice structure if the electron density is lower than a critical value. This is because the potential energy dominates the kinetic energy at low densities and consequently the electrons arrange into a crystal structure in order to minimize the potential energy. This theory has an important dimensionless parameter, the Wigner– Seitz radius at zero temperature, which can be viewed as an estimation of the average electron spacing. For two-dimensional one component plasma, the correlation energy and chemical potential of each ion are

Here R is the radius of a Wigner– Seitz crystal with its cell approximated as a disk (Fig. 4).

Based on the proposed correlation energy and chemical potential, one important conclusion is the prediction of charge inversion.^{[20, 44]} In electrophoresis experiments, charge inversion is often shown to reverse complex mobility. Section 8 will illustrate how this theory can be used to explain DNA reentrant condensation behavior.

Though the counterion electrostatic correlation theory can explain the attraction between like-charge polyelectrolytes such as DNA, it cannot explain cation specificity. For example, it is has been shown that transition metal ions (Mn²⁺ and Cd²⁺ ) cause guanine-cytosine (GC) rich DNA condensation despite their low affinity for phosphate, whereas alkali earth metal ions (Mg²⁺ , Ca²⁺ ) cannot condense DNA in spite of their high affinity for phosphate. To explain this, Kornyshev and Leikin suggested that details of the surface charge pattern may determine the specificity and energetics of DNA condensation. They proposed an electrostatic interaction theory to explain cation specificity in DNA condensation.^[47] Counterion condensation in helix grooves results in axial charge separation, which enables the attraction between opposite charges along the DNA– DNA contact and forms an electrostatic zipper-like structure. This theory also uses the counterion condensation concept; counterion condensation on the phosphate strand is thought to reduce the attraction because of weaker charge separation, which is consistent with the observation that Mg²⁺ and Ca²⁺ cannot induce DNA condensation despite their high affinity for phosphate. In contrast, the counterion condensation of transition metal ions, such as Mn²⁺ , onto the major groove of the DNA leads to attraction between DNA regions.

The advantage of the electrostatic zipper theory is that it considers the helical structure of the DNA molecule and the different ion binding patterns on DNA. However, similar to other theories, the ion size effect is not included and hence it may be difficult to use the theory to explain DNA condensation caused by ions with larger sizes.

The hydration force concept was developed based on measurements of the osmotic pressure as a function of the distance between the polymers. It was found that when the distance between parallel aligned DNA molecules is 5 Å – 15 Å , ion strength does not have a significant effect on the interaction force; the repulsion pressure decays exponentially with a characteristic decay length of 2.5 Å – 3.5 Å .^[57]

The precise physical mechanism of hydration force remains unclear. It is assumed that the force originates from the reconfiguration of water molecules near condensed polymers such as DNA.^{[58, 59]} As dipoles, water molecules orderly arrange near DNA because of its electric field. The order degree decays exponentially with the distance from DNA. Leikin et al. suggested that the magnitude of the hydration force is related to the order degree of the water molecules on the surface.^[45] Attraction is due to complementary arrangement and repulsion results from symmetrical arrangement. It is difficult to determine the magnitude of the force since it is the sum of weak interactions between numerous molecules. In spite of this, a number of general conclusions regarding the order parameter formalism have been developed. Consider two similar, homogenous planar surfaces separated by distance h. The repulsion pressure is

If the surfaces are complementary, the pressure is attractive with a magnitude of

Here R and A are coefficients, λ _W is the water correlation length (approximately 4 Å – 5 Å . If the surfaces have a lattice structure with periodicity a, homogeneous pressure is replaced by inhomogeneous pressure:

where σ is the surface charge density. For B-DNA, a = 34 Å , λ _W = 4 Å – 5 Å . The effective correlation length is

with a value of 1.6 Å ∼ 1.8 Å , which is comparable with experimental measurements.

The drawback of the hydration force theory is that it is an experiment-oriented theory, which can only explain the resisting and attracting forces involved in DNA condensation; the physical elements remain unclear and untested.

Consider a macromolecule consisting of N monomer segments with a persistent length L_p and width d. The free energy of each macromolecule can be expressed as a function of the swelling coefficient α (= R/R₀):

where R and R₀ are the gyration radius of the real chain and the Gaussian chain, B and C are the second and third virial coefficients. The first term on the right side of the equation describes elastic free energy and the second and third terms are the free energy of interaction. Minimizing the free energy F over α results in the following equation:

where . The parameter χ characterizes the solvent quality and y describes chain stiffness. For very stiff chains (L_p/d ≫ 1) and a given χ , α has two stable solutions (α _c and α _g; α _c ≪ α _g) and one unstable solution. The solutions signify that the coil and globule conformations coexist in a region with finite parameter width 0 < χ _l < χ < χ _h, where χ _l and χ _h are the lower and upper limits of the parameter. It can be concluded that the width of the coexisting region decreases with increasing numbers of chain segments. In other words, the coexistence of the coil and globule conformations is caused by the finite length of the macromolecule.

The coil– globule transition theory successfully explains experimental observations of the coexistence phenomenon. Specifically, single-molecule fluorescence experiments have shown that the coil conformation coexists with the globule conformation.^{[60, 61]} However, it should be noted that the coil– globule theory is in the structure of the theory proposed by Lifschitz, ^[62] which states that the virial expansion only applies to low DNA segment concentrations. Since the concentration in DNA condensates can be quite high and the polyelectrolyte nature of DNA is not directly reflected in this theory, more detailed studies are not possible.

The phenomenon of cooperative binding is a common concept in biochemistry; ^[66] when one ligand molecule binds to a macromolecule, the binding affinity of other ligand molecules may be strengthened or weakened. In 1910, Hill proposed an equation to quantify this effect in order to explain the sigmoidal shape of a curve estimating hemoglobin oxygen saturation as a function of the partial pressure. The equation is:

where θ is fractional occupancy (the ratio of the number of bound sites to the number of total sites), [L] denotes ligand concentration, K_d denotes association constant, and n is the Hill coefficient describing cooperativity. If n = 1, the binding is independent of the concentration of the bound ligands. If n > 1, the binding shows positive cooperativity, whereas cooperativity is negative if n < 1. Using this equation, Hill confirmed that the binding of oxygen molecules to hemoglobin is cooperative and that the Hill coefficient is 2.8∼ 3.

DNA condensation can be viewed as a process of ligand molecules (e.g. multivalent ions or proteins) binding to receptor molecules (DNA). If the binding is cooperative, it should be observed in single-molecule DNA condensation experiments. Consider a suspending DNA molecule with an initial end-to-end length L₀ micromanipulated by either the MT (magnetic tweezers) or OT (optical tweezers) method. When ligand molecules with a cooperative effect are added, it can be assumed that the cooperative effect is proportional to the number of bound molecules. Binding leads to the DNA length L gradually shortening. Therefore, the number of bound molecules is assumed to be proportional to L₀ − L. At the same time, the shortening of DNA length results in a decrease of the effective interaction volume, which is proportional to L − L_f. Here L_f is the final length of the saturated DNA. It is assumed that within the corresponding volume, ligand molecules cannot bind to the DNA. Combining these two terms yields an equation describing the DNA condensation rate:

where K = k · C_p · k₀, k is reaction constant, C_p is ligand concentration, k₀ is the number of ligands to condense DNA per unit length. Solving the equation gives a length-time curve with a sigmoidal shape (Fig. 5), which fits well with experimental measurements.^[67] That is to say, cooperative behavior is reflected in the sigmoidal shape of the length-time curve of single-molecule DNA condensation measurements. Hence the old concept of “ cooperativity” is reinterpreted in the new single-molecule experiment.

Fig. 5.

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	Fig. 5. Comparison between the DNA condensation curve and the cooperative behavior model.[67]

Resolubilization is another concept developed through ensemble experiments. Pelta et al. studied polyamine and cobalthexamine-induced DNA condensation by measuring the amount of uncondensed DNA in the supernatant.^[17] They found that the addition of multivalent cations leads initially to DNA precipitation; further addition leads to the resolubilization of the DNA condensates, which is confirmed by the increased DNA concentration in the supernatant.

Resolubilzation is manifested as reentrant condensation behavior (Fig. 6) in single-molecule experiments. Besteman et al. studied charge inversion accompanied by DNA condensation caused by multivalent ions using MT.^[68] They initially used the DLS method to measure the mobility of condensed DNA particles. A change in mobility from negative to positive indicates charge inversion. When the concentration of spermine in 1-mM Tris buffer is greater than 0.5 mM, mobility becomes positive. Increasing the Tris concentration to 10-mM hinders charge inversion, inducing the charge-inversion concentration of spermine to increase to 1 mM. The addition of 50-mM KCl salt leads to the disappearance of charge inversion at spermine concentrations < 3 mM. Next, they used MT experiments to elucidate how charge inversion hinders DNA condensation. The measurements of the condensation force (F_c) as a function of spermine concentration (c) shows that the force increases to a maximum at c = 10^{− 3} M– 10^{− 2} M, followed by a gradual decrease in F_c. When c > 0.5 M, condensation ceases. The decrease in F_c indicates reentrant condensation at the single-molecule level. The coincidence of the measured and extrapolated charge inversion concentrations and the peak concentrations of the condensation force measurements suggests that charge inversion results in peak F_c, consistent with the prediction of the counterion correlation theory. The applied theory of this research was proposed by Nguyen et al.; ^[44] the main point of the theory is that when a polyelectrolyte is neutralized to carry sufficient low effective charges, the electrostatic correlation overcomes the electrostatic repulsion resulting in condensation. However, when the effective charge of the polyelectrolyte is reversed, electrostatic repulsion becomes dominant again and reentrant condensation occurs. This is another example of an old concept represented as a new phenomenon in single-molecule experiments.

Fig. 6.

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	Fig. 6. Reentrant condensation behavior.

Todd et al. found a similar phenomenon in multivalent ion-induced DNA condensation and provided an alternate explanation for it.^{[69, 70]} Combining MT and osmotic stress on DNA assemblies, they separated the attractive and repulsive components of the interaction force inside condensed DNA. Two invariant properties of multivalent cation-mediated DNA interactions were identified. At force-balance equilibrium, repulsive forces decay exponentially with a cation-independent 2.3 Å ± 0.1 Å decay length; the decay length of the attractive forces is constantly 2.3± 0.2 times higher than the repulsive component. These decay lengths were then used to test current theories. The counterion correlation theory predicts that the ratio of attractive force to repulsive force is between 1.2– 1.8, which is inconsistent with the measurements. The Debye– Hü ckel interaction and order parameter formalism for hydration forces theories predict the ratio is approximately 2, which agrees well with the data.

Since the counterion correlation theory is not consistent with experimental results, how can reentrant condensation behavior be explained? Todd et al. noticed that reentrant condensation occurs near the solubility limit of the multivalent cation salt. Near this limit, lower-valent anion-associated multivalent cations are likely to form, which increase the energetic cost for neutralizing DNA. For example, trivalent cations associate with monovalent anions and form divalent ion pairs. Divalent ion pairs have lower condensing abilities than multivalent ions, therefore it is thought that high concentrations of multivalent cations lead to decreased DNA condensation force. Based on this, they successfully used the traditional Manning theory to explain their data.

These two groups have provided different explanations for the same phenomenon based on their measurements and analysis. According to the counterion correlation theory, charge inversion always leads to reentrant condensation, but is reentrant condensation necessarily caused by charge inversion? Several experimental points need to be taken into consideration in order to answer this question. Firstly, charge inversion does exist in DNA condensation; secondly, the critical concentrations at which charge inversion occurs are not highly agreeable with the critical reentrant condensation concentration; thirdly, in order to apply their model, Todd et al. used the simplification that the DNA condensate is neutral at high multivalent ion concentrations; and lastly, the experimental conditions of the two studies were dissimilar. Based on these facts, it is best to be cautious in reaching a final conclusion until further experimental verification is obtained.

We recently reported that this reentrant behavior also occurs in the Ψ condensation.^[71] The MT measurements show that by increasing Mg²⁺ concentration, the DNA condensation in polyethylene glycol (PEG) solution undergoes a reentrant process, which was verified by the decreased critical condensation force. The critical concentration of Mg²⁺ at which reentrant condensation occurred in the study was as low as ∼ 3 mM (in 30% PEG 6000 solution). Such an observation seems to rule out the possibility of the ion pair formation, because Mg²⁺ has lower affinity to anions compared with the cases of multi-valent cations, and consequently the ion pair effect is negligible in the studied concentration range. Nevertheless, the difference between the two similar reentrant behaviors in the two different scenarios (i.e., the Ψ condensation and the multi-valent cation-induced DNA condensation) remains to be investigated further. In spite of this, the predictions of the theories can be either proved or disproved by the above-mentioned single molecule experiments. Therefore, additional and similar experiments should be conducted to thoroughly evaluate the various theories.