Comparison of hard-cylinder and screened Coulomb interactions in the modeling of supercoiled DNAs

Counterion condensation and shape within Poisson-Boltzmann theory

An analytical approximation to the nonlinear Poisson-Boltzmann (PB) equation is applied to charged macromolecules that possess one-dimensional symmetry and can be modeled by a plane, infinite cylinder, or sphere. A functional substitution allows the nonlinear PB equation subject to linear boundary conditions to be transformed into an approximate linear (Debye-Hückel-type) equation subject to nonlinear boundary conditions. A simple analytical result for the surface potential of such polyelectrolytes follows, leading to expressions for the amount of condensed (or renormalized) charge and the electrostatic Helmholtz energy for polyelectrolytes. Analytical high-charge/low-salt and low-charge/high-salt limits are shown to be similar to results obtained by others based on PB or counterion condensation theory. Several important general observations concerning polyelectrolytes treated within the context of PB theory can be made including: (1) all charged surfaces display some counterion condensation for finite electrolyte concentration, (2) the effect of surface geometry is described primarily by the sum of the Debye constant and the mean curvature of the surface, (3) two surfaces with the same surface charge density and mean curvature condense approximately identical fractions of counterions, (4) the amount of condensation is not determined by a predefined "condensation distance" although such a distance can be determined uniquely from it, and (5) substantial condensation occurs if the Debye constant of the electrolyte is much less than the mean curvature of a highly charged polyelectrolyte.

Estimation of the persistence length of DNA from the torsion elastic constant and supercoiling free energy: effect of ethylene glycol

Two different methods are proposed to estimate the persistence length ( P) of DNA from the measured torsion elastic constant (alpha) and the twist energy parameter ( E T ) that governs the supercoiling free energy. The first method involves Monte Carlo simulations and reversible-work calculations of E T for model DNAs that possess the measured alpha and selected trial values of P. Comparison of the computed E T values with the experimental value allows estimation of P (or equivalently the bending elastic constant (kappa beta)) by interpolation. A far simpler, though less accurate, alternative is to solve a previously conjectured analytical relation connecting E T , alpha, kappa beta (or P), and an unknown "constant" ( B). The present simulations are used to ascertain the optimum value of B and to assess the validity and accuracy of that relation. Within the simulation errors, P values obtained from the measured alpha and E T via this analytical expression agree with those determined from the simulations and E T values reckoned from the input alpha and kappa beta by this analytical expression agree with the corresponding simulated values. Although B is found to be insensitive to variation in alpha, it appears to decline slightly with increasing kappa beta. The original analytical expression is modified to take this apparent variation of B with kappa beta into account. By using this modified analytical relation to estimate P (from the measured alpha and E T ) or E T (from the input alpha and kappa beta), much closer agreement is obtained respectively with the values of P or E T obtained from the simulations. As specific examples, these methods are applied to determine P in 0 and 20 w/v % ethylene glycol, which has been shown to induce a structural transition in duplex DNA.

Computational modeling of the chromatin fiber

The packing of the genomic DNA in the living cell is essential for its biological function. While individual aspects of the genome architecture, such as DNA and nucleosome structure or the arrangement of chromosome territories are well studied, much information is missing for a unified description of cellular DNA at all its structural levels. Computer modeling can contribute to such a description. We present here some typical approaches to models of the chromatin fiber, including different amounts of detail in the description of the local nucleosome structure. The main results from our simulations are that the physical properties of the chromatin fiber can be well described by a simplified model consisting of cylinder-like nucleosomes connected by flexible DNA segments, with a geometry determined by the bending and twisting angles between nucleosomes. Randomness in the local geometry - such as random absence of linker histone H1 - leads to a dramatic increase in the chromatin fiber flexibility. Furthermore, we show that chromatin is much more flexible to bending than to stretching, and that the structure of the chromatin fiber favors the formation of sharp bends.

Torsional rigidities of weakly strained DNAs

Measurements on unstrained linear and weakly strained large (> or =340 bp) circular DNAs yield torsional rigidities in the range C = 170-230 fJ fm. However, larger values, in the range C = 270-420 fJ fm, are typically obtained from measurements on sufficiently small (< or =247 bp) circular DNAs, and values in the range C = 300-450 fJ fm are obtained from experiments on linear DNAs under tension. A new method is proposed to estimate torsional rigidities of weakly supercoiled circular DNAs. Monte Carlo simulations of the supercoiling free energies of solution DNAs, and also of the structures of surface-confined supercoiled plasmids, were performed using different trial values of C. The results are compared with experimental measurements of the twist energy parameter, E(T), that governs the supercoiling free energy, and also with atomic force microscopy images of surface-confined plasmids. The results clearly demonstrate that C-values in the range 170-230 fJ fm are compatible with experimental observations, whereas values in the range C > or = 269 fJ fm, are incompatible with those same measurements. These results strongly suggest that the secondary structure of DNA is altered by either sufficient coherent bending strain or sufficient tension so as to enhance its torsional rigidity.

Two-angle model and phase diagram for chromatin

We have studied the phase diagram for chromatin within the framework of the two-angle model. Only a rough estimation of the forbidden surface of the phase diagram for chromatin was given in a previous work of Schiessel. We revealed the fine structure of this excluded-volume borderline numerically and analytically. Furthermore, we investigated the Coulomb repulsion of the DNA linkers to compare it with the previous results.

Polymer chain models of DNA and chromatin

Many properties of the genome in the cell nucleus can be understood by modeling DNA and chromatin as a flexible polymer chain. This article introduces into current models for such a coarse-grained description and reviews some recent results from our own group. Examples given are the unrolling of DNA from the histone core and the response of the 30 nm chromatin fiber to mechanical stretching.

Parallel, quantitative measurement of protein binding to a 120-element double-stranded DNA array in real time using surface plasmon resonance microscopy

Quantitative, real-time measurement of kinetics of sequence-specific binding of DNA-binding proteins to double-stranded DNA (dsDNA) immobilized in a 10 x 12 array on a planar gold surface is demonstrated using surface plasmon resonance (SPR) microscopy. This binding of the yeast transcription factor Gal4 to a 120-spot dsDNA array made with alternating 200-microm spots of its dsDNA operator sequence and an unrelated DNA sequence proves that this method could be used to simultaneously monitor the kinetics of binding of proteins to 120 different dsDNA sequences with a sensitivity to approximately 0.5 pg (<2 x 10(7) molecules) of bound protein in each array spot at a time resolution of 1 s. The method is label free and also allows absolute quantitative determination of the binding stoichiometry (i.e., the number of proteins bound per dsDNA) at each time. The dsDNA array was fabricated using a robotic microspotting system to deliver nanoliter droplets of biotinylated dsDNA solutions onto a streptavidin linker layer immobilized with biotinylated alkylthiols on a thin gold film. Simultaneous monitoring of binding to the many array elements allows the use of reference spots (i.e., array elements with unrelated dsDNA sequences) to correct the signal for nonspecific protein-DNA binding and changes in bulk refractive index of the solutions in the SPR microscope's flow cell. This allows high-throughput analyses of the kinetics and equilibrium of protein-dsDNA binding.

Monte Carlo simulations of locally melted supercoiled DNAs in 20 mM ionic strength

Mesoscopic models of unmelted and locally melted supercoiled DNAs in 20 mM ionic strength are simulated over a range of linking difference from deltal = 0 to -26 turns, or superhelix density from sigma = 0 to -0.062. A domain containing m = 0, 28, or 56 melted basepairs (out of 4349 total) is modeled simply by a region of suitable length with substantially reduced torsion and bending elastic constants. Average structural properties are calculated from the saved configurations, and a reversible work protocol is used to calculate the supercoiling free energy, The cross-writhe between duplex and melted regions (defined herein) is found to be negligibly small. The total writhe, radius of gyration, and ordered elements of the diagonalized inertial tensor are found to be nearly universal functions of the residual linking difference (deltal(r)) associated with the duplex region, independent of m. However, deformability of the tertiary structure, as manifested by the variance of those same properties, is not a universal function of deltal(r)), but depends upon m.delta (SC) varies with deltal(r)) more strongly than deltal(r)) (2)due to the low ionic strength. The twist energy parameter, E (T) obtained from the simulated delta G(SC), deltal(r)), and net twisting strain of the melted region T (D), is found to be independent of m, hence also of the torsion and bending elastic constants of the melted region. However, E(T) increases linearly with -deltalr), which leads to 1). a small overestimation of E (T) for any given deltal(r)) when E(T) is determined from the observed deltal and deltal (r) by the protocol of Bauer and Benham; and 2). a significant enhancement of the apparent slope, -dE(T)/d(T), obtained via the protocol of Bauer and Benham, relative to the actual slope at fixed delta l(r). After taking these two effects into account, the theoretical and experimental values E(T) and -dE(T)/d(T) values agree rather well. For the larger deltal the melted regions are found preferentially in the linker domains between interwound arms, rather than in the apical regions at the ends of interwound arms.

Microspotting Streptavidin and Double-Stranded DNA Arrays on Gold for High-Throughput Studies of Protein−DNA Interactions by Surface Plasmon Resonance Microscopy

We present two strategies for microspotting 10 x 12 arrays of double-stranded DNAs (dsDNAs) onto a gold-coated glass slide for high-throughput studies of protein-DNA interactions by surface plasmon resonance (SPR) microscopy. Both methods use streptavidin (SA) as a linker layer between a biotin-containing mixed self-assembled monolayer (SAM) and biotinylated dsDNAs to produce arrays with high packing density. The primary mixed SAM is produced from biotin- and oligo(ethylene glycol)-terminated thiols bonded as thiolates onto the gold surface. In the first method, a robotic microspotter is used to deliver nanoliter droplets of dsDNA solution onto a uniform layer of this SA ( approximately 2 x 10(12) SA/cm(2)). SPR microscopy shows a density of (5-6) x 10(11) dsDNA/cm(2) (0.2-0.3 dsDNA/SA) in the array elements. The second method uses instead a microspotted array of this SA linker layer, onto which the microspots of dsDNA are added with spatial registry. SPR microscopy before addition of the dsDNA shows a SA coverage of 2 x 10(12) SA/cm(2) within the spots and a dsDNA density of 8.5 +/- 3.5 x 10(11) dsDNA/cm(2) (0.3-0.7 dsDNA/SA, depending on the length of dsDNA) after dsDNA spotting. We demonstrate the ability to simultaneously monitor protein binding with the SPR microscope in many 200-microm spots with 1-s time resolution and sensitivity to <1 pg of protein.

Nonmonotonic variation with salt concentration of the second virial coefficient in protein solutions

The osmotic virial coefficient B2 of globular protein solutions is calculated as a function of added salt concentration at fixed pH by computer simulations of the "primitive model." The salt and counterions as well as a discrete charge pattern on the protein surface are explicitly incorporated. For parameters roughly corresponding to lysozyme, we find that B2 first decreases with added salt concentration up to a threshold concentration, then increases to a maximum, and then decreases again upon further raising the ionic strength. Our studies demonstrate that the existence of a discrete charge pattern on the protein surface profoundly influences the effective interactions and that linear and nonlinear Poisson Boltzmann theories fail for large ionic strength. The observed nonmonotonicity of B2 is compared with experiments. Implications for protein crystallization are discussed.

Extensions of counterion condensation theory. I. Alternative geometries and finite salt concentration

The counterion condensation theory originally proposed by Manning is extended to take account of both finite counterion concentration (m(C)) and the actual structure of the array of discrete changes. Counterion condensation is treated here as a binding isotherm problem, in which the unknown free volume is replaced by an unknown local binding constant beta', which is expected to vary with m(C) and polyion structure. The relation between the condensed fraction of counterion charge, r, beta' and m(C) is obtained from the relevant grand partition function via the maximum term method. In the case of the single polyion in a large salt reservoir, the result is practically identical to Manning's equation. In order to determine the values of beta' and r at arbitrary m(C), a second relation between r, beta' and m(C) is required. We propose an alternative auxiliary relation that is equivalent to previous assumptions near m(C) = 0, but which yields qualitatively correct and quantitatively useful results at finite m(C). Simple expressions for r vs. m(C) and beta' vs. m(C) are obtained by simultaneously solving the binding isotherm and auxiliary equations. Then r and beta' are evaluated for five different linear arrays of infinite extent with different geometries: (1) a straight line of charges with uniform axial spacing; (2) two parallel lines of in-phase uniformly spaced charges; (3) a single-helix of discrete charges with uniform axial spacing; (4) a double-helix of discrete charges with uniform axial spacing of pairs of charges; (5) a cylindrical array of many parallel charged lines, chosen to simulate a uniformly charged cylinder. In all cases, the computed binding isotherms exhibit qualitatively correct behavior. As m(C) approaches zero, r approaches the Manning limit, r = 1-1/(L(B)/b) where b is the average axial spacing of electronic charges in the array and L(B) is the Bjerrum length. However, beta' varies with polyion geometry, even in the zero salt limit, and matches the Manning value only in the case of a single straight charged line. With increasing m(C), r declines significantly below its limiting value whenever lambda(b) > or approximately equal 0.3, where lambda is the Debye screening parameter. In the case of cylindrical arrays containing either 2 or 100 parallel charged lines, r also decreases, whenever lambda(d) > or approximately equal 2.0, where d is the diameter of the array. In the case of two parallel charged lines, each with axial charge spacing b=3.4 A, which are separated by d = 200 A, r exhibits a plateau value, 0.76, characteristic of the two combined lines, when lambda(d)<<2.0, and declines with increasing m(C) to a shelf value, 0.52, characteristic of either single line, when lambda(d) > or approximately equal 2.0 and the lines become effectively screened from one another. beta' behaves in a roughly similar fashion. In the case of a cylindrical array of charged lines with the diameter and linear charge density of DNA, the r-values predicted by the present theory agree fairly well with those predicted by non-linear Poisson-Boltzmann theory up to 0.15 M uni-univalent salt.

Monte Carlo simulations of supercoiled DNAs confined to a plane

Recent advances in atomic force microscopy (AFM) have enabled researchers to obtain images of supercoiled DNAs deposited on mica surfaces in buffered aqueous milieux. Confining a supercoiled DNA to a plane greatly restricts its configurational freedom, and could conceivably alter certain structural properties, such as its twist and writhe. A program that was originally written to perform Monte Carlo simulations of supercoiled DNAs in solution was modified to include a surface potential. This potential flattens the DNAs to simulate the effect of deposition on a surface. We have simulated transfers of a 3760-basepair supercoiled DNA from solution to a surface in both 161 and 10 mM ionic strength. In both cases, the geometric and thermodynamic properties of the supercoiled DNAs on the surface differ significantly from the corresponding quantities in solution. At 161 mM ionic strength, the writhe/twist ratio is 1.20-1.33 times larger for DNAs on the surface than for DNAs in solution and significant differences in the radii of gyration are also observed. Simulated surface structures in 161 mM ionic strength closely resemble those observed by AFM. Simulated surface structures in 10 mM ionic strength are similar to a minority of the structures observed by AFM, but differ from the majority of such structures for unknown reasons. In 161 mM ionic strength, the internal energy (excluding the surface potential) decreases substantially as the DNA is confined to the surface. Evidently, supercoiled DNAs in solution are typically deformed farther from the minimum energy configuration than are the corresponding surface-confined DNAs. Nevertheless, the work (Delta A(int)) done on the internal coordinates, which include uniform rotations at constant configuration, during the transfer is positive and 2.6-fold larger than the decrease in internal energy. The corresponding entropy change is negative, and its contribution to Delta A(int) is positive and exceeds the decrease in internal energy by 3.6 fold. The work done on the internal coordinates during the solution-to-surface transfer is directed primarily toward reducing their entropy. Evidently, the number of configurations available to the more deformed solution DNA is vastly greater than for the less deformed surface-confined DNA.

Stretching single-stranded DNA: interplay of electrostatic, base-pairing, and base-pair stacking interactions

Recent single-macromolecule observations revealed that the force/extension characteristics of single-stranded DNA (ssDNA) are closely related to solution ionic concentration and DNA sequence composition. To understand this, we studied the elastic property of ssDNA through the Monte Carlo implementation of a modified freely jointed chain (FJC), with electrostatic, base-pairing, and base-pair stacking interactions all incorporated. The simulated force-extension profiles for both random and designed sequences have attained quantitative agreements with the experimental data. In low-salt solution, electrostatic interaction dominates, and at low forces, the molecule can be more easily aligned than an unmodified FJC. In high-salt solution, secondary hairpin structure appears in ssDNA by the formation of base pairs between complementary bases, and external stretching causes a hairpin-coil structural transition, which is continuous for ssDNA made of random sequences. In designed sequences such as poly(dA-dT) and poly(dG-dC), the stacking potential between base pairs encourages the aggregation of base pairs into bulk hairpins and makes the hairpin-coil transition a discontinuous (first-order) process. The sensitivity of elongation to the base-pairing rule is also investigated. The comparison of modeling calculations and the experimental data suggests that the base pairing of single-stranded polynucleotide molecules tends to form a nested and independent planar hairpin structure rather than a random intersecting pattern.

Effect of polyethylene glycol on the supercoiling free energy of DNA

The supercoiling free energy of pUC19 DNA [2686 base pairs (bp)] was measured in various concentrations of PEG 8000 (polyethylene glycol; molecular weight 8000) by the topoisomer distribution method. The effective twist energy parameter (E(T)) that governs the supercoiling free energy declined linearly by 1.9-fold with increasing w/v % PEG from 0 to 7.5%, which lies below the threshold for intermolecular condensation. In principle, PEG could affect E(T) either via an osmotic exclusion mechanism or by altering the torsion elastic constant, bending rigidity, or self-repulsions of the DNA. Possible alterations of the DNA secondary structure and torsion elastic constant were assessed by CD spectroscopy and time-resolved fluorescence polarization anisotropy of intercalated ethidium. Up to 7.5% PEG, the secondary structure of the DNA remained largely unaltered, as evidenced by (1) the absence of any significant change in the CD spectrum, (2) an extremely small relative decrease (-0.0013) in intrinsic twist, and (3) a negligibly small change in the torsion elastic constant. The observed reduction in E(T) cannot be ascribed primarily to a decrease in torsion elastic constant, and most likely does not stem from a decrease in bending rigidity either. The decrease in medium dielectric constant due to PEG should increase the self-repulsions, and thereby increase E(T), which is opposite to the observed trend. Instead, the observed decline in E(T) is attributed to an osmotic exclusion mechanism. The change in molar volume excluded to the PEG (Delta V(ex)), when the linking difference converts from Delta l = 0 to Delta l = +/-1, was determined from the observed E(T) value and PEG osmotic pressure at each concentration. The experimental Delta V(ex) values agree well with theoretical estimates reckoned for a simple osmotic exclusion model, in which PEG is excluded by hard-core interactions from a concentric cylindrical volume around every duplex segment. The difference in volume excluded to PEG between the Delta l = 0 and the Delta l = +/-1 topoisomers is attributed entirely to the approximately 0.7 additional writhe "crossing" of two duplex strands at roughly 90 degrees, which is known to occur in the latter species. When the separation between the duplex centers at the "crossing" was adjusted so that the theoretical estimate of Delta V(ex) matched the experimental value at each PEG concentration, a value near 5.7 nm was obtained in each case. The invariance and plausible magnitude of this mean separation at the crossing provide strong support for this simple osmotic exclusion model. An alternative model, in which the PEG is excluded from the entire coil envelope of the DNA out to its radius of gyration, perhaps because it decreases the local dielectric constant, was also considered. The estimated difference in excluded volume in that case exceeds the experimental value by a factor of nearly 10(4), and could be ruled out on that basis.

Effective interaction between helical biomolecules

The effective interaction between two parallel strands of helical biomolecules, such as deoxyribose nucleic acids (DNA), is calculated using computer simulations of the "primitive" model of electrolytes. In particular we study a simple model for B-DNA incorporating explicitly its charge pattern as a double-helix structure. The effective force and the effective torque exerted onto the molecules depend on the central distance and on the relative orientation. The contributions of nonlinear screening by monovalent counterions to these forces and torques are analyzed and calculated for different salt concentrations. As a result, we find that the sign of the force depends sensitively on the relative orientation. For intermolecular distances smaller than 6 A it can be both attractive and repulsive. Furthermore, we report a nonmonotonic behavior of the effective force for increasing salt concentration. Both features cannot be described within linear screening theories. For large distances, on the other hand, the results agree with linear screening theories provided the charge of the biomolecules is suitably renormalized.

Mapping the bacterial cell architecture into the chromosome

A genome is not a simple collection of genes. We propose here that it can be viewed as being organized as a 'celluloculus' similar to the homunculus of preformists, but pertaining to the category of programmes (or algorithms) rather than to that of architectures or structures: a significant correlation exists between the distribution of genes along the chromosome and the physical architecture of the cell. We review here data supporting this observation, stressing physical constraints operating on the cell's architecture and dynamics, and their consequences in terms of gene and genome structure. If such a correlation exists, it derives from some selection pressure: simple and general physical principles acting at the level of the cell structure are discussed. As a first case in point we see the piling up of planar modules as a stable, entropy-driven, architectural principle that could be at the root of the coupling between the architecture of the cell and the location of genes at specific places in the chromosome. We propose that the specific organization of certain genes whose products have a general tendency to form easily planar modules is a general motor for architectural organization in the bacterial cell. A second mechanism, operating at the transcription level, is described that could account for the efficient building up of complex structures. As an organizing principle we suggest that exploration by biological polymers of the vast space of possible conformation states is constrained by anchoring points. In particular, we suggest that transcription does not always allow the 5'-end of the transcript to go free and explore the many conformations available, but that, in many cases, it remains linked to the transcribing RNA polymerase complex in such a way that loops of RNA, rather than threads with a free end, explore the surrounding medium. In bacteria, extension of the loops throughout the cytoplasm would therefore be mediated by the de novo synthesis of ribosomes in growing cells. Termination of transcription and mRNA turnover would accordingly be expected to be controlled by sequence features at both the 3'- and 5'-ends of the molecule. These concepts are discussed taking into account in vitro analysis of genome sequences and experimental data about cell compartmentalization, mRNA folding and turnover, as well as known structural features of protein and membrane complexes.

Electrostatic-undulatory theory of plectonemically supercoiled DNA

We present an analytical calculation of the electrostatic interaction in a plectonemic supercoil within the Poisson-Boltzmann approximation. Undulations of the supercoil strands arising from thermal motion couple nonlinearly with the electrostatic interaction, giving rise to a strong enhancement of the bare interaction. In the limit of fairly tight winding, the free energy of a plectonemic supercoil may be split into an elastic contribution containing the bending and torsional energies and an electrostatic-undulatory free energy. The total free energy of the supercoil is minimized according to an iterative scheme, which utilizes the special symmetry inherent in the usual elastic free energy of the plectoneme. The superhelical radius, opening angle, and undulation amplitudes in the radius and pitch are obtained as a function of the specific linking difference and the concentration of monovalent salt. Our results compare favorably with the experimental values for these parameters of Boles et al. (1990. J. Mol. Biol. 213:931-951). In particular, we confirm the experimental observation that the writhe is a virtually constant fraction of the excess linking number over a wide range of superhelical densities. Another important prediction is the ionic strength dependence of the plectonemic parameters, which is in reasonable agreement with the results from computer simulations.