ABSTRACT In this paper we introduce an important parameter called the iso-competition point (ICP), to characterize the competition binding to DNA in a two-cation-species system. By imposing the condition of charge neutralization fraction equivalence theta^sub 1^ = Ztheta^sub z^ upon the two simultaneous equations in Manning's counterion condensation theory, the ICPs can be calculated. Each ICP, which refers to a particular multivalent concentration where the charge fraction on DNA neutralized from monovalent cations equals that from the multivalent cations, corresponds to a specific ionic strength condition. At fixed ionic strength, the total DNA charge neutralization fractions theta^sub ICP^ are equal, no matter whether the higher valence cation is divalent, trivalent, or tetravalent. The ionic strength effect on ICP can be expressed by a semiquantitative equation as ICP^sub za^/ICP^sub zb^ = (I^sub a^/I^sub b^)^sup Z^, where I^sub a^, I^sub b^ refers to the instance of ionic strengths and Z indicates the valence. The ICP can be used to interpret and characterize the ionic strength, valence, and DNA length effects on the counterion competition binding in a two-species system. Data from our previous investigations involving binding of Mg^sup 2+^, Ca^sup 2+^, and Co(NH^sub 3^)^sup 3+^^sub 6^ to lambda-DNA-HindIII fragments ranging from 2.0 to 23.1 kbp was used to investigate the applicability of ICP to describe counterion binding. It will be shown that the ICP parameter presents a prospective picture of the counterion competition binding to polyelectrolyte DNA under a specific ion environment condition.
INTRODUCTION
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 (Mg^sup 2+^, Ca^sup 2+^), and trivalent cations (hexamine cobalt (III) and spermidine^sup 3+^) with lambda-DNA-HindIII fragments ranging from 2,027 to 23,130 bp in Tris-borateEDTA 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 theta, 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
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.
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^sub a^ value was corrected iteratively using the Davies equation (Perrin and Dempsey, 1979) to be pK'^sub 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-Huckel screening parameter kappa should be calculated according to the known ionic strength (Li et al., 1996). The condensation volumes V^sub p1^, V^sub p2^, V^sub p3^, and V^sub 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 theta^sub I_ICP^ and theta^sub Z_ICP^ and the critical multivalent cation concentration ICP were obtained. At this particular ICP we have the following relationship: oI_ICP = Ztheta^sub Z_ICP^, which states the concept of the ICP mathematically. It illustrates that the DNA charge neutralized by the monovalent cation theta^sub 1_ICP^ is equal to that neutralized by the higher valence cation, which is Ztheta^sub Z_ICP^.
PROPERTIES of ICP
In this section we present and discuss important features of the calculated ICP values to have an essential understanding of the nature of ICP.
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 C^sub 1^, where ionic strength equals C^sub 1^, 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 = C^sub 1^. 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, theta^sub ICP^, has a higher value, which is closer to the critical charge neutralization fraction of 0.890.
INTERPRETATION OF COUNTERION BINDING BY ICP
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 theta^sub 1^, from multivalent theta^sub 2^, (theta^sub 3^) and the total theta versus the logarithm of multivalent ion concentration C^sub 2^ (C^sub 3^) 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(NH^sub 3^)^sup 3+^^sub 6^ to lamda-DNA-HindIII fragments in 22.79 mM ionic strength and 19.80 mM monovalent ion concentration; another is the binding of Mg^sup 2+^ to A-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 (0.01400 (mu)M) 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 (mu)M, where trivalent and monovalent cations have equal charge neutralization fractions. Under these conditions the trivalent cation concentration 0.387 (mu)M is nothing but the ICP. After this point the trivalent cation dominates the binding competition. In the case of divalent cations (0.01-400 (mu)M) 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 theta^sub 2^ is much slower than theta^sub 3^ in the previous case, and the same is true of the decreased rate of theta^sub 1^ lowering. The divalent cation concentration (ICP) corresponding to the crossover point is 53.70 (mu)M 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.
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 theta^sub 1^, theta^sub 2^ and theta 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 [Mg^sup 2+^] (0.01-400 (mu)M) competing with monovalent cation [Na+, Tris+] at concentrations of 8.65 mM, 17.67 mM, and 29.73 mM binding to lambda-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 theta^sub 1^ is equal to that from divalent 2theta^sub 2^, shifts to the right when ionic strength increases. The ICP values, where divalent cation concentration C^sub 2^ corresponds to the crossover point values, are 12.98, 53.70, and 150 (mu)M 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^sub za^/ICP^sub Zb^ = (la/IIb)z, where Z = 2, to test the above data. Using our ICP data we have 53.70 (mu)M/12.98 (mu)M = 4.14, and for ionic strength (17.70 mM/8.67 mM)^sup 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^sub Z^ = Const I^sup 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.
DISCUSSION
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 theta^sub 1^ = Ztheta^sub 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 theta^sub 1^ neutralized by monovalent cations. The red portion of the rectangle refers to the charge neutralization fraction 2theta^sub 2^, and the total volume of the rectangle indicates the total charge neutralization fraction theta. 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 = C^sub 1^, covers the practical range of 1-30 mM, while at each ionic strength, Ca^sup 2+^ varies over the range of 0.1 to 300 (mu)M. 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 Ca^sup 2+^ 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 theta^sub 1^ curve is represented by the green rectangle; the 2theta^sub 2^ curve represented by the red rectangle; and the theta curve is represented by the entire icon including green and red rectangles, which slowly increases with the increase in Ca^sup 2+^
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[Author Affiliation]
Anzhi Z. Li and Kenneth A. Marx
Department of Chemistry, University of Massachusetts Lowell, Lowell, Massachusetts 01854 USA
[Author Affiliation]
Received for publication 9 November 1998 and in final form 16 March 1999.
Address reprint requests to Dr. Kenneth A. Marx, Department of Chemistry, University of Massachusetts Lowell, One University Ave., Lowell, MA 01854. Tel.: 978-934-3658; Fax: 978-934-3013; E-mail: Kenneth_Marx@uml.edu.
Dr. Li's present address is Genome Therapeutics Corporation, 100 Beaver Street, Waltham, MA 02453. E-mail: anzhi.li@genomecorp.com.
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