INTRODUCTION

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

For over two decades, the phenomenon of counterion condensation has attracted many scientists' experimental and theoretical attention, either from a biological view or polyelectrolyte perspective. Particularly from the standpoint of conformational properties of polyion DNA, such as the helix-coil transition (Widom and Baldwin, 1980; Bloomfield, 1991), the condensation based collapse of DNA and its resulting structure (Allison et al., 1981; Marx and Ruben, 1983, 1986; Marx and Reynolds, 1982, 1989; Arscott et al., 1990; Plum et al., 1990; Li et al., 1992), has been fairly well studied. A variety of experimental approaches, including NMR (Granot and Kearns, 1982), differential scanning calorimetry (Labarbe et al., 1996), Raman spectroscopy (Langlais et al., 1990), absorption measurements (Manzini et al., 1990), electrophoretic light scattering (Rhee and Ware, 1983; Xia et al., 1993), and gel electrophoresis (Ma and Bloomfield, 1995; Li et al., 1996, 1998) have been employed to measure the counterion binding to DNA. These studies compared the experimental results with predictions from polyelectrolyte theory, either Manning's counterion condensation (CC) theory (Manning, 1977, 1978, 1981), and/or the Poisson-Boltzmann (PB) equation.

Our previous studies were focused on the counterion competition binding of multivalent versus monovalent counterions onto polyelectrolyte DNA. The interactions of divalent cations (Mg²⁺, Ca²⁺), and trivalent cations (hexamine cobalt (III) and spermidine³⁺) with -DNA-HindIII fragments ranging from 2,027 to 23,130 bp in Tris-borate-EDTA buffer solutions were examined using pulsed gel electrophoresis (Li et al., 1996, 1998; Holzwarth et al., 1989). The divalent or trivalent counterions competed with Tris⁺ and Na⁺ for binding onto polyion DNA, and the competition binding details were investigated by measuring the reduction of DNA gel electrophoretic mobility under a specific ion environment. The measured data were interpreted by the Henry gel model (Cantor and Schimmel, 1980; Rice and Nagasawa, 1961) and Manning's CC theory (Manning, 1977, 1978). Good agreement was found between the experimental data, based on mobility reduction measurements converted to the total charge neutralization fraction , and the predicted value from Manning's CC theory.

In our studies of counterion competition binding, the ionic strength, counterion valence, and DNA molecular weight effects on the competition binding were carefully investigated (Li et al., 1996, 1997, 1998). From these studies we developed an insight into the counterion binding system which revealed that the above phenomena could all be associated with an important parameter defined to be the iso-competition point [ICP] (Li et al., 1997). The ICP refers to a critical multivalent cation concentration, at a given ionic strength and temperature, where the multivalent cations possess a charge neutralization fraction on DNA equal to that of monovalent cations. In the following paper we discuss the definition and calculation of ICP, and how ICP may be applied to characterize and interpret the counterion competition binding. It will be shown that the ICP parameter actually presents a prospective picture of the counterion competition binding to polyelectrolyte DNA under a specific ion environment condition.

	DEFINITION AND COMPUTATION

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

In this section we define ICP through three simultaneous equations that include Manning's two equations, and present the approach to calculate ICP corresponding to a specific ion environment.

Definition of ICP

The parameter ICP is closely associated with Manning's CC theory. In Manning's two-variable CC system, two species of counterions of different valences are present in solution to compete for binding to the polyion. Suppose in the solution the lower-valence cation is monovalent (the general case), and the higher-valence cations are one of the following: divalent, trivalent or tetravalent. In the above competition environment, ₁ represents the fraction of charge neutralized by monovalent, and _Z is the number of Z-valent ions condensed per phosphate where the valence Z = 2, 3, or 4, respectively. Z_Z refers to the fraction of DNA charge neutralized by the divalent, trivalent, or tetravalent cations. To calculate the iso-competition point, another equation, Eq. 3, is added to Manning's original two equations (Manning, 1978; Wilson and Bloomfield, 1979). The three simultaneous equations are as follows:

(1)

(2)

(3)

where C₁ and C_Z represent the molar concentration of monovalent and higher valence counterions. Similarly, V_p1 and V_pZ represent the volume per mole phosphate where the condensed counterions (monovalent or higher valent) are considered to be territorially bound. The formula for calculating V_pZ is given in our previous publication (Li et al., 1998). The charge density parameter, , is an important parameter in Manning's counterion condensation theory (Manning, 1978) which governs the counterion binding. The term b refers to the average axial charge spacing. Specifically, b = 1.7 Å and  = 4.2 at 25°C for double-stranded DNA in aqueous solution.  is the Debye-Hückel screening parameter, which is itself dependent on the ionic strength (Cantor and Schimmel, 1980). The constant e refers to the base of natural logarithms.

Solving simultaneous Eqs. 1 and 2 iteratively, the charge neutralization fractions ₁, and _Z, then the total charge neutralization fraction

(4)

can be obtained based on the following specific conditions of ion environment: the Debye-Hückel screening parameter  (corresponding to a specific ionic strength) and the molar concentrations of competing cations, C₁ and C_Z. In general, C_Z C₁, and the ionic strength contribution from the higher valence cation can be ignored relative to the ionic strength contribution from the monovalent, and thus the ionic strength of the system is close to the monovalent concentration C₁. For simplifying the ICP calculation, we assume the ionic strength remains constant when low concentrations of multivalent cation C_Z (C_Z/C₁ < 0.01) are added to the two-species system because its contribution to the ionic strength can be ignored. For a fixed ionic strength and C₁, each multivalent concentration C_Z corresponds to a pair of predicted charge neutralization fraction values ₁ and _Z, and then total charge neutralization fraction  based on Eq. 4.

Unlike the case of solving Manning's two variable equations, the higher valence cation concentration C_Z becomes a variable to be determined, instead of a known one. Only one pair of charge neutralization fraction data (_{1_ICP} and _{Z_ICP}) among all infinite pairs (₁ and _Z) will be selected to satisfy Eq. 3 when the ionic strength C₁ and temperature are fixed. The multivalent cation concentration C_{Z_ICP} along with _{1_ICP} and _{Z_ICP} are the three solutions for the simultaneous Eqs. 1-3 corresponding to a fixed ionic strength C₁ and temperature. C_{Z_ICP} is defined to be the iso-competition point (ICP). At this critical multivalent cation concentration, ICP, the two rival monovalent, and multivalent cation competitors possess an equal binding fraction of the polyion DNA charge.

Computation of ICP

To obtain the value of ICP where Eqs. 1-3 need to be solved simultaneously, the MATHEMATICA tool (Wolfram, 1991) was employed to execute the iterative numerical calculations, and the computation approach is similar to that described in the following publications (Li et al., 1996, 1998). The main procedure is divided into two steps: calculation of the ion environment and simultaneous solution of the three equations.

If the calculation of ICP is associated with a particular experimental environment, it is necessary in the first step to analyze the ion environment and calculate the correct ionic strength and monovalent cation concentration as well. The ionic strength is calculated corresponding to a particular ion environment based on the Henderson-Hasselbalch equation (Perrin and Dempsey, 1979) where the pK_a value was corrected iteratively using the Davies equation (Perrin and Dempsey, 1979) to be pK'_a, corresponding to the chosen ionic strength. If ICP calculation is not associated with a real experimental system, but is a simulation, the first step could be skipped. In the simulation system, the ionic strength value could be set equal to the monovalent concentration.

For the second step, obtaining the numerical solutions of the three simultaneous equations, the Debye-Hückel screening parameter  should be calculated according to the known ionic strength (Li et al., 1996). The condensation volumes V_p1, V_p2, V_p3, and V_p4 then need to be computed corresponding to the individual valences Z = 1, 2, 3, 4, respectively (Li et al., 1998). With all parameters substituted in Eqs. 1-3, these simultaneous equations are solved iteratively by a small program based on the MATHEMATICA tool (Wolfram, 1991). The specific charge neutralization fractions _{1_ICP} and _{Z_ICP} and the critical multivalent cation concentration ICP were obtained. At this particular ICP we have the following relationship: _{1_ICP} = Z_{Z_ICP}, which states the concept of the ICP mathematically. It illustrates that the DNA charge neutralized by the monovalent cation _{1_ICP} is equal to that neutralized by the higher valence cation, which is Z_{Z_ICP}.

	PROPERTIES of ICP

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

In this section we present and discuss important features of the calculated ICP values to have an essential understanding of the nature of ICP.

Iso-charge neutralization line

Fig. 1 maps the logarithm (ICP) versus total charge neutralization fraction _ICP. Three curves, representing the respective divalent, trivalent, and tetravalent cases, where the monovalent counterion competes with the higher valence ions (Z = 2, 3, 4), are presented. When calculating the ICPs in Fig. 1, the ionic strength was assumed to be equal to the monovalent concentration C₁, and the temperature was set to 23°C. The ionic strength ranges from 1 to 75 mM, which covers a wide range of ionic strengths appropriate to practical applications. Meanwhile, the ICP range is 0.175 µM to 0.95 mM for divalent, 5.08 × 10⁵ to 20 µM for trivalent, and 2.3 × 10⁷ to 0.5 µM for tetravalent. The horizontal lines drawn in Fig. 1 are iso-charge neutralization lines. Each line connects the ICPs for divalent, trivalent, and tetravalent cations versus total charge neutralization fraction. The series of iso-charge neutralization lines presented from bottom to top corresponds to different ionic strengths in the range 1-75 mM, respectively. It is clear from these data that when cation valence changes, ICP values change dramatically, but the total charge neutralization fraction _ICP remains constant for a specific ionic strength condition. However, the iso-charge neutralization lines demonstrate that regardless of cation valence, when the competition reaches the equivalence point (ICP), where the charge neutralization fraction is equal from higher valence and monovalent cations, the total charge neutralization fraction _ICP is constant for a specific ion environment.

View larger version (27K): [in this window] [in a new window]   FIGURE 1   Logarithm of ICP versus total charge neutralization . Three sets of points from right to left represent the ICPdi, ICPtri, and ICPtetra, respectively. The horizontal lines are iso-charge neutralization lines corresponding to a specific ionic strength. The range of ionic strength represented by points from bottom to top is 1-75 mM.

The iso-charge neutralization line can be proved theoretically. Based on Eqs. 3 and 4, we have:

(5A)

(5B)

(5C)

where _ICP,di, _ICP,tri, and _ICP,tetra refer to the total charge neutralization at ICP where the multivalent cation is divalent, trivalent, and tetravalent, respectively. The right sides 2 _{1_ICP} in Eqs. 5, A-C are equal since ionic strengths are the same. The iso-charge neutralization equation then could be expressed as:

(5D)

Notice that the total charge neutralization _ICP in Fig. 1 slowly rises with increasing ionic strength.

Semi-quantitative equations

ICP values change when valence or ionic strength changes. In Fig. 1, ICP values decrease dramatically when valence increases, and the relationship ICP_di ICP_tri ICP_tetra is always observed. For example, in a two-species system of 20 mM ionic strength, ICP_di equals 68.81 µM for divalent cations, ICP_tri is 0.39 µM for trivalents, and ICP_tetra is 2.755 × 10³ µM for tetravalents. It is not surprising to see the ICP change so dramatically versus valence if one considers Manning's two-variable CC theory. As we might expect, this ICP valence phenomenon is somewhat similar to the valence effect of CC theory.

Below we present the semiquantitative equations to describe the ICP value dependence on two variables, valence Z and ionic strength I, based on a curve-fitting approach. Fig. 2 shows the relationship between ICP and ionic strength, for monovalent counterions competing with divalent cations in Fig. 2 A, trivalent cations in Fig. 2 B, and tetravalent cations in Fig. 2 C. The three best-fit equations obtained from the curve-fitting shown in the figures can be written in a generalized form as:

(6A)

and

(6B)

View larger version (20K): [in this window] [in a new window]   FIGURE 2   Relationship of ICP versus ionic strength where the higher-valence cation is: (A) divalent, (B) trivalent, and (C) tetravalent, respectively. Curve fits through the calculated points are of the type described in each panel.

ICP and CCP

In studying the condensation-based collapse of DNA, it is well known that reaching a critical charge neutralization fraction (0.890) of the DNA is required to bring about the DNA collapse (Wilson and Bloomfield, 1979). The critical collapse conditions were described (Li et al., 1996) by C₁, where ionic strength equals C₁, and the critical collapse point (CCP) defined as the trivalent cation concentration. Fig. 3 A presents curves of ICP and CCP versus ionic strength, where I = C₁. At a fixed temperature, each ionic strength has an ICP value, where the counterion competition binding reaches a balance and the charge neutralization fraction is equal from the competing monovalent and trivalent counterions. Also, each ionic strength has a CCP, where the total charge neutralization fraction is 0.890. It is clear that at any ionic strength the CCP value is much higher than ICP. That is because ICP is a transition point where the trivalent counterions start to dominate the binding to DNA, and CCP is the "final" point where the charge neutralization fraction caused mainly by trivalent counterions finally brings about the conformational collapse of DNA. In Fig. 3 B the nonlinear relationship between CCP and ICP is observed. The slope of the curve is lower when ionic strength increases in Fig. 3 B, which corresponds to the decreasing distance between ICP and CCP points when ionic strength rises in Fig. 3 A. This is the case because at higher ionic strength the total charge neutralization, _ICP, has a higher value, which is closer to the critical charge neutralization fraction of 0.890. 

View larger version (25K): [in this window] [in a new window]   FIGURE 3   (A) Logarithm of ICP and CCP versus ionic strength. CCP is the critical collapse point, corresponding to a total charge neutralization of DNA of 0.890, which will bring about the tertiary structure collapse of DNA to insoluble toroidal self-assembled structures. (B) Relationship between the logarithm of CCP and ICP.

	INTERPRETATION OF COUNTERION BINDING BY ICP

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

The concept of ICP is closely associated with Manning's two-variable CC theory, and it is introduced to characterize and interpret the counterion competition binding in the two-species system.

ICP and valence effects

In Fig. 4 the charge neutralization fraction from monovalent ₁, from multivalent ₂, (₃) and the total  versus the logarithm of multivalent ion concentration C₂ (C₃) is presented. Notice that the heavy symbols represent the trivalent case and the light symbols represent the divalent case. The data of Fig. 4 were calculated by CC theory to correspond to two separate competition binding systems we have experimentally investigated. One is the binding of Co(NH₃)₆³⁺ to -DNA-HindIII fragments in 22.79 mM ionic strength and 19.80 mM monovalent ion concentration; another is the binding of Mg²⁺ to -DNA-HindIII fragments in 17.70 mM ionic strength and 17.67 mM monovalent ion concentration. The theoretical curves show the competition binding between divalent and trivalent cations with monovalent cations directly. Upon inspection of trivalent cations (0.01-400 µM) competing with monovalent cations (19.80 mM) in an ionic strength of 22.79 mM, one notices that the monovalent charge fraction drops rapidly, whereas the trivalent cation charge fraction rises at the same rate, and the two curves cross at 0.387 µM, where trivalent and monovalent cations have equal charge neutralization fractions. Under these conditions the trivalent cation concentration 0.387 µM is nothing but the ICP. After this point the trivalent cation dominates the binding competition. In the case of divalent cations (0.01-400 µM) competing with monovalent cations (17.67 mM) in an ionic strength of 17.70 mM, a different quantitative binding behavior is observed. The rising rate of charge neutralization fraction ₂ is much slower than ₃ in the previous case, and the same is true of the decreased rate of ₁ lowering. The divalent cation concentration (ICP) corresponding to the crossover point is 53.70 µM where divalent and monovalent cations have equal charge neutralization fractions. Notice that the ICP of divalent cations is much larger (more than two orders of magnitude) than ICP of trivalent cations under very similar ion environment conditions. Knowing the values of ICP (divalent and trivalent), one can evaluate how rapidly the trivalent cation will dominate the DNA binding competition in contrast to the much less effective divalent cation competitor. The valence effect on competition binding reflected by ICP here is consistent with the valence behavior of ICP shown in Fig. 1. It is clear that the ICP parameter provides an important reference point for viewing a competition binding system, and indeed these data may help to design binding experiments.

View larger version (32K): [in this window] [in a new window]   FIGURE 4   Valence effect on DNA charge neutralization fractions 1, 22, (33), and , calculated by CC theory versus logarithm of divalent or trivalent cation concentration under similar ionic conditions. For trivalent cations the ionic strength is 22.79 mM and C1 = 19.80 mM, while for divalent cations the ionic strength is 17.70 mM and C1 = 17.67 mM. Note that the light symbols indicate the divalent versus monovalent competition system, and the heavy symbols indicate the trivalent versus monovalent competition system.

ICP and ionic strength effect

The ionic strength effect on counterion binding has been discussed thoroughly in previous publications (Li et al., 1996, 1998). Here we intend to characterize the ionic strength effect on counterion binding using the novel parameter ICP. Fig. 5 presents charge neutralization fraction ₁, ₂ and  versus the logarithm of divalent cation concentration at three different ionic strengths that correspond to experimental data from Li et al., 1998. The theoretical curves were calculated by CC theory. The competition conditions in Fig. 5, A-C, are divalent cations [Mg²⁺] (0.01-400 µM) competing with monovalent cation [Na⁺, Tris⁺] at concentrations of 8.65 mM, 17.67 mM, and 29.73 mM binding to -DNA-HindIII fragments at ionic strengths of 8.67 mM, 17.70 mM, and 29.78 mM, respectively. It is observed that the crossover point, where charge neutralization from monovalent cation ₁ is equal to that from divalent 2₂, shifts to the right when ionic strength increases. The ICP values, where divalent cation concentration C₂ corresponds to the crossover point values, are 12.98, 53.70, and 150 µM in Fig. 5, A-C, respectively. The above ICP values characterize the ionic strength effect. The higher the ionic strength, the larger the ICP, which indicates that a higher divalent cation concentration is required to reach the point where it can start to dominate the binding. Quantitatively, one can use Eq. 6 B: ICP_Za/ICP_Zb = (I_a/I_b)^Z, where Z = 2, to test the above data. Using our ICP data we have 53.70 µM/12.98 µM = 4.14, and for ionic strength (17.70 mM/8.67 mM)² = 4.16. The small difference in these two values may be caused by using limited significant digits in these calculations or it may be due to the necessity for another constant added to Eq. 6 A, as ICP_Z = Const_Z I^Z + Const. Nonetheless, Eqs. 6 A and B are useful to predict an unknown ICP from a given ICP and the ratio of the known ionic strengths.

View larger version (39K): [in this window] [in a new window]   FIGURE 5   Ionic strength effect on charge neutralization fractions 1, 22, and , calculated by CC theory. The 1, 22, and  versus logarithm of divalent cation concentration is presented under the following ionic conditions: (A) 8.67 mM ionic strength and 8.65 mM monovalent ion concentration; (B) 17.70 mM ionic strength and 17.67 mM monovalent ion concentration; (C) 29.78 mM ionic strength and 29.73 mM monovalent ion concentration. Curves represented by solid circle, open circle, and diamond symbols indicate the charge neutralization fraction 1, 22, and , respectively.

ICP and DNA size effect

Previous investigations (Li et al., 1997, 1998) reveal that the experimental data collected by pulsed gel electrophoresis shows a distribution of normalized mobility µ/µ_o or converted charge neutralization fraction  over the range of DNA lengths from 2.0 to 23.1 kbp. The larger the fragment length, the higher the total charge neutralization fraction. It was observed that the distribution of (µ/µ_o) was dependent on the ionic strength, cation valence, and multivalent cation concentration. Fig. 6 presents the normalized mobility µ/µ_o versus divalent Ca²⁺ concentration at different ionic strengths (I = 4.84, 15.0, 20.2, and 25.4 mM). The agreement between experimental measurements (symbols) and CC prediction (solid line) is good. Also, a consistent DNA length effect on µ/µ_o associated with the ionic strength was observed. The relative shift (µ/µ_o)/(µ/µ_o) was calculated by subtracting the µ/µ_o of the largest fragment (f₁) from that of the smallest (f₆) and normalizing the difference by the µ/µ_o value of f₆. The (µ/µ_o)/(µ/µ_o) "shift" increased with increasing C₂ and decreasing ionic strength. It is obvious to see that the shift phenomenon can be ignored in Fig. 6 D, which corresponds to a higher ionic strength, while the shift phenomenon is strongest in Fig. 6 A, which corresponds to the lowest ionic strength. The study of Li and Marx, 1997 demonstrates that the shift phenomenon only occurs when divalent concentration C₂ is high enough to dominate the competition binding, which is measured by comparing C₂ to the ICP, and ICP itself is ionic strength-dependent.

View larger version (27K): [in this window] [in a new window]   FIGURE 6   Normalized mobility µ/µ0 of -DNA-HindIII fragments versus Ca2+ concentration under the following ionic conditions: (A) 4.84 mM ionic strength and 4.83 mM monovalent ion concentration; (B) 14.97 mM ionic strength and 14.95 mM monovalent ion concentration; (C) 20.18 mM ionic strength and 20.14 mM monovalent ion concentration; (D) 25.43 mM ionic strength and 25.38 mM monovalent ion concentration. The experimental data for different molecular weight bands represented by the symbols were fit by Manning's CC theory, shown by the solid lines.

Fig. 7 A is a survey plot that includes four columns: ionic strength (I), ICP, Ca²⁺ concentration, and shift (µ/µ_o)/(µ/µ_o). The three factors (I, ICP, Ca²⁺ concentration) govern the shift (µ/µ_o)/(µ/µ_o) magnitude. The value of each variable is represented by the width of the corresponding rectangle. The first column shows four ionic strengths, the second column shows four ICP values corresponding to the four ionic strengths. In the third column, the concentration of Ca²⁺ was represented by three rectangles: 10, 20, 40 µM for each ionic strength and ICP values. The fourth column shows the measured relative shift (µ/µ_o)/(µ/µ_o) associated with the particular horizontal row of I, ICP, and Ca²⁺ conditions. ICP is the key to the connection between measured shift (µ/µ_o)/(µ/µ_o) and the controlling factors (I, ICP, Ca²⁺ concentration) in Fig. 7 A. When the Ca²⁺ concentration is larger than or close to its ICP where the divalent cations dominate the competition binding, the shift (µ/µ_o)/(µ/µ_o), which is a function of DNA length and the ion environment, is measurable. Because the divalent counterion condensation brings about a conformation change of the DNA fragments, this could result in an end-to-end distance decrease that is length-dependent (Li et al., 1997). For example, in the case of 4.8 mM ionic strength, the Ca²⁺ concentrations (10, 20, 40 µM) are all larger than ICP (4.1 µM) and the shifts are all observable. Obviously, when the Ca²⁺ concentration equals 40 µM, the maximum shift is shown both in Fig. 7 A and Fig. 6 A. By contrast, when the ionic strength equals 25.4 mM and the ICP is high (111.3 µM), the divalent Ca²⁺ concentrations (10, 20, 40 µM) are all smaller than ICP and the shift (1%) magnitude can be ignored.

View larger version (30K): [in this window] [in a new window]   FIGURE 7   Visualizations of competition binding and ICP. (A) Survey plot of the interrelationship among ionic strength I, ICP, Ca2+ concentration, and relative mobility shift (µ/µo)/(µ/µo). Ionic strength, ICP, Ca2+ concentration, and measured shift values (µ/µo)/(µ/µo) are "listed" in four columns. The value of each parameter is represented by the width of the corresponding rectangle. It is shown that the occurrence of a significant shift is dependent on whether the ratio of Ca2+ concentration to ICP is larger than or close to 1 for the solution conditions corresponding to any horizontal row of the table. (B) Icon graph mapping competition binding and iso-competition points. The logarithm of divalent cation concentration (x axis) and logarithm of ionic strength (y axis) locates the icon in the graph. A rectangular icon is used to present three parameters by its color and volume. The green portion of the rectangle refers to the charge fraction 1 neutralized by monovalent cations, the red portion of the rectangle refers to the charge fraction 22 neutralized by divalent cations, and the total volume of the rectangle indicates the total charge neutralization fraction . The round-edged rectangular icon is used to present charge fractions of ICP whose green and red charge fraction portions are exactly equal (1 = 22). Notice that at one ionic strength (one horizontal row), only one ICP exists.

	DISCUSSION

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

Visualizing competition binding and ICP

In this paper we introduce an important parameter, the iso-competition point (ICP), to characterize the competition binding in a two-species system. By imposing the condition of charge neutralization equivalence ₁ = Z_Z upon Manning's two simultaneous equations, ICPs can be calculated, each corresponding to a specific ionic strength. With the help of Fig. 7 B we review the ICP concept and its connection with Manning's CC theory in a visual way.

Fig. 7 B is an icon graph (Pickett and Grinstein, 1988) drawn using visualization techniques (Nielson et al., 1997) and Java programing (Campione and Walrath, 1997). Instead of a point located in a coordinate system in the traditional scatter plot, an icon is used to present multiple variables in a 2-D plot. A rectangular icon is chosen to present three variables by color and volume. The green portion of each rectangle refers to the charge fraction ₁ neutralized by monovalent cations. The red portion of the rectangle refers to the charge neutralization fraction 2₂, and the total volume of the rectangle indicates the total charge neutralization fraction . A logarithm coordinate system was chosen to locate the icons in the ion environment comprised of the ionic strength (y axis) and divalent cation concentration (x axis). The ionic strength, where I = C₁, covers the practical range of 1-30 mM, while at each ionic strength, Ca²⁺ varies over the range of 0.1 to 300 µM. At a given ionic strength, upon scanning the graph from left to right, it is clear that the green portion of successive icons is gradually decreasing, while the red portion is increasing with rising divalent cation concentration. There is a special icon whose shape is rounded at the edges and its green portion exactly equals the red portion. This special icon signifies the charge neutralization fraction of ICP and the value of ICP at a particular divalent cation concentration. Notice that for one ionic strength, only one ICP exists and the special icon shows the ion competition balance visually. All the icons located on the left side of the ICP icon have larger green portions than red ones, while all the icons located on the right side of the ICP icon have larger red portions than green ones. It is clearly shown that the ICP represents a transition point, after which the divalent cations dominate the counterion binding, and the charge on polyion DNA is mostly neutralized by the divalent cations. Upon viewing the graph from bottom to top and from left to right simultaneously, the ionic strength effect will be clear. The icons in the bottom row with the lowest ionic strength (1 mM) have green portions decreasing rapidly with the increase of divalent cation concentration. It reveals at the low ionic strength that divalent cations strongly compete with the monovalent cations. Even at very low Ca²⁺ concentration they dominate the competition binding, and the position of the ICP icon is located at low divalent cation concentration. For the top row with the highest ionic strength (30 mM) the competition picture is reversed, and the ICP icon appears at very high cation concentration. When viewing the icons by rows, 12 "curves" are shown. Three curves can be viewed in each horizontal row corresponding to one ionic strength. The ₁ curve is represented by the green rectangle; the 2₂ curve represented by the red rectangle; and the  curve is represented by the entire icon including green and red rectangles, which slowly increases with the increase in Ca²⁺.

	SUMMARY

TOP ABSTRACT INTRODUCTION DEFINITION AND COMPUTATION PROPERTIES of ICP INTERPRETATION OF COUNTERION... DISCUSSION SUMMARY REFERENCES

The important points are summarized as follows:

1.	Under fixed ionic strength condition (assume monovalent C1 = I), only one ICP exists where the charge neutralization fraction on DNA from monovalent cations equals that from multivalent cations. That is, 1_ICP = ZZ_ICP;
2.	For fixed ionic strength, the total charge neutralization fractions ICP are the same at ICP, no matter whether the higher valence cation is divalent, trivalent, or tetravalent. That is, ICP,di = ICP,tri = ICP,tetra = ICP;
3.	The ionic strength effect on ICP could be expressed as ICPZa/ICPZb = (Ia/Ib)Z; this relationship can be viewed in the first and second column in Fig. 7 A, and the numbers fit the equation nicely;
4.	The valence effect on ICP is very strong, but so far it hasn't been quantitatively expressed;
5.	ICP is able to interpret and characterize the ionic strength, valence, and DNA length effects of the counterion competition binding in a two-species system;
6.	Since we are discussing counterion condensation, where only territorially bound counterions fit Manning's CC theory, it is only these ions for which the ICP concept is relevant. For example, the transition metal cations such as Cu2+ and Zn2+ are known to bind strongly to the DNA nitrogenous bases rather than phosphate groups; they should not be treated using the ICP concept (Li et al., 1998).