David Swigon

Theory of sequence-dependent DNA elasticity

The elastic properties of a molecule of duplex DNA are strongly dependent on nucleotide sequence. In the theory developed here the contribution ψn of the nth base-pair step to the elastic energy is assumed to be given by a function ψn of six kinematical variables, called tilt, roll, twist, shift, slide, and rise, that describe the relative orientation and displacement of the nth and (n+1)th base pairs. The sequence dependence of elastic properties is determined when one specifies the way ψn depends on the nucleotides of the two base pairs of the nth step. Among the items discussed are the symmetry relations imposed on ψn by the complementarity of bases, i.e., of A to T and C to G, the antiparallel nature of the DNA sugar–phosphate chains, and the requirement that ψn be independent of the choice of the direction of increasing n. Variational equations of mechanical equilibrium are here derived without special assumptions about the form of the functions ψn, and numerical solutions of those equations are...

Theory of Supercoiled Elastic Rings with Self-Contact and Its Application to DNA Plasmids

Methods are presented for obtaining exact analytical representations of supercoiled equilibrium configurations of impenetrable elastic rods of circular cross-section that have been pretwisted and closed to form rings, and a discussion is given of applications in the theory of the elastic rod model for DNA. When, as here, self-contact is taken into account, and the rod is assumed to be inextensible, intrinsically straight, transversely isotropic, and homogeneous, the important parameters in the theory are the excess link Δℒ (a measure of the amount the rod was twisted before its ends were joined), the ratio ω of the coefficients of torsional and flexural rigidity, and the ratio d of cross-sectional diameter to the length of the axial curve C. Solutions of the equations of equilibrium are given for cases in which self-contact occurs at isolated points and along intervals. Bifurcation diagrams are presented as graphs of Δℒ versus the writhe of C and are employed for analysis of the stability of equilibrium configurations. It is shown that, in addition to primary, secondary, and tertiary branches that arise by successive bifurcations from the trivial branch made up of configurations for which the axial curve is a circle, there are families of equilibrium configurations that are isolas in the sense that they are not connected to bifurcation branches by paths of equilibrium configurations compatible with the assumed impenetrability of the rod. Each of the isolas found to date is connected to a bifurcation branch by a path which, although made up of solutions of the governing equations, contains regions on which the condition of impenetrability does not hold.

The Elastic Rod Model for DNA and Its Application to the Tertiary Structure of DNA Minicircles in Mononucleosomes

Explicit solutions to the equations of equilibrium in the theory of the elastic rod model for DNA are employed to develop a procedure for finding the configuration that minimizes the elastic energy of a minicircle in a mononucleosome with specified values of the minicircle size N in base pairs, the extent w of wrapping of DNA about the histone core particle, the helical repeat h(0)b of the bound DNA, and the linking number Lk of the minicircle. The procedure permits a determination of the set Y(N, w, h(0)b) of integral values of Lk for which the minimum energy configuration does not involve self-contact, and graphs of writhe versus w are presented for such values of Lk. For the range of N of interest here, 330 < N < 370, the set Y(N, w, h(0)b) is of primary importance: when Lk is not in Y(N, w, h(0)b), the configurations compatible with Lk have elastic energies high enough to preclude the occurrence of an observable concentration of topoisomer Lk in an equilibrium distribution of topoisomers. Equilibrium distributions of Lk, calculated by setting differences in the free energy of the extranucleosomal loop equal to differences in equilibrium elastic energy, are found to be very close to Gaussian when computed under the assumption that w is fixed, but far from Gaussian when it is assumed that w fluctuates between two values. The theoretical results given suggest a method by which one may calculate DNA-histone binding energies from measured equilibrium distributions of Lk.

Continuum Model of Collective Cell Migration in Wound Healing and Colony Expansion

Collective cell migration plays an important role during wound healing and embryo development. Although the exact mechanisms that coordinate such migration are still unknown, experimental studies of moving cell layers have shown that the primary interactions governing the motion of the layer are the force of lamellipodia, the adhesion of cells to the substrate, and the adhesion of cells to each other. Here, we derive a two-dimensional continuum mechanical model of cell-layer migration that is based on a novel assumption of elastic deformation of the layer and incorporates basic mechanical interactions of cells as well as cell proliferation and apoptosis. The evolution equations are solved numerically using a level set method. The model successfully reproduces data from two types of experiments: 1), the contraction of an enterocyte cell layer during wound healing; and 2), the expansion of a radially symmetric colony of MDCK cells, both in the edge migration velocity and in cell-layer density. In accord with experimental observations, and in contrast to reaction-diffusion models, this model predicts a partial wound closure if lamellipod formation is inhibited at the wound edge and gives implications of the effect of spatially restricted proliferation.

Relating Single-Molecule Measurements to Thermodynamics

Measurements made on large ensembles of molecules are routinely interpreted using thermodynamics, but the normal rules of thermodynamics may not apply to measurements made on single molecules. Using a polymer stretching experiment as an example, it is shown that in the limit of a single, short molecule the outcome of experimental measurements may depend on which variables are held fixed and which are allowed to fluctuate. Thus an experiment in which the end-to-end distance of the polymer molecule is fixed and the tension fluctuates yields a different result than an experiment where the force is fixed and the end-to-end distance fluctuates. It is further shown that this difference is due to asymmetry in the distribution of end-to-end distances for a single molecule, and that the difference vanishes in the appropriate thermodynamic limit; that is, as the polymer molecule becomes long compared to its persistence length. Despite these differences, much of the thermodynamic formalism still applies on the single-molecule level if the thermodynamic free energies are replaced with appropriate potentials of mean force. The primary remaining differences are consequences of the fact that unlike the free energies, the potentials of mean force are not in general homogeneous functions of their variables. The basic thermodynamic concepts of an intensive or extensive quantity, and the thermodynamic relationships that follow from them, are therefore less useful for interpreting single-molecule experiments.

Theory of the influence of end conditions on self‐contact in DNA loops

Explicit solutions of the equations of Kirchhoff’s theory of elastic rods are employed to derive properties of the tertiary structure of a looped segment of DNA that is subject to geometric constraints imposed at its end points by bound proteins. In appropriate circumstances small changes in such boundary data cause a nearly planar loop to undergo a continuous and reversible transition that can be described as a 180° rotation taking the loop from an uncrossed to a singly crossed structure in which sequentially separated base pairs are brought into proximity. Expressions are derived relating points and angles of crossing to end conditions, and results are presented that facilitate the calculation of changes in elastic energy during such transitions.

Implications of the dependence of the elastic properties of DNA on nucleotide sequence

Recent advances in structural biochemistry have provided evidence that not only the geometric properties but also the elastic moduli of duplex DNA are strongly dependent on nucleotide sequence in a way that is not accounted for by classical rod models of the Kirchhoff type. A theory of sequence–dependent DNA elasticity is employed here to calculate the dependence of the equilibrium configurations of circular DNA on the binding of ligands that can induce changes in intrinsic twist at a single base–pair step. Calculations are presented of the influence on configurations of the assumed values and distribution along the DNA of intrinsic roll and twist and a modulus coupling roll to twist. Among the results obtained are the following. For minicircles formed from intrinsically straight DNA, the distribution of roll–twist coupling strongly affects the dependence of the total elastic energy Ψ on the amount α of imposed untwisting, and that dependence can be far from quadratic. (In fact, for a periodic distribution of roll–twist coupling with a period equal to the intrinsic helical repeat length, Ψ can be essentially independent of α for –90° < α <90°.) When the minicircle is homogeneous and without roll–twist coupling, but with uniform positive intrinsic roll, the point at which Ψ attains its minimum value shifts towards negative values of α. It is remarked that there are cases in which one can relate graphs of Ψ versus α to the ‘effective values’ of bending and twisting moduli and helical repeat length obtained from measurements of equilibrium distributions of topoisomers and probabilities of ring closure. For a minicircle formed from DNA that has an ‘S’ shape when stress–free, the graphs of Ψ versus α have maxima at α = 0. As the binding of a twisting agent to such a minicircle results in a net decrease in Ψ, the affinity of the twisting agent for binding to the minicircle is greater than its affinity for binding to unconstrained DNA with the same sequence.

A simple mathematical model of signaling resulting from the binding of lipopolysaccharide with Toll-like receptor 4 demonstrates inherent preconditioning behavior

The complex biology of Gram-negative bacterial lipopolysaccharide (LPS) is central to the acute inflammatory response in sepsis and related diseases. Repeated treatment with LPS can lead to desensitization or enhancement of subsequent responses both in vitro and in vivo (a phenomenon known as preconditioning). Previous computational studies have demonstrated a role for anti-inflammatory influences in this process (J. Day, J. Rubin, Y. Vodovotz, C.C. Chow, A. Reynolds, G. Clermont, A reduced mathematical model of the acute inflammatory response: II. Capturing scenarios of repeated endotoxin administration. J. Theor. Biol. 242 (2006) 237). Since LPS signals via Toll-like receptor 4 (TLR4), we created a simple mathematical model in order to address the role of this receptor in both the normal and preconditioned response to LPS. We created a non-linear system of ordinary differential equations, consisting of free LPS, free TLR4, bound complex LPS-TLR4, and an intracellular signaling cascade (lumped into a single variable). We simulate the effects of preconditioning by small and large repeated doses of LPS on the system, varying the timing of the doses as well as the rate of expression of TLR4. Our simulations suggest that a simplified model of LPS/TLR4 signaling can account for complex preconditioning phenomena without invoking a specific signaling inhibition mechanism, but rather based on the dynamics of the signaling response itself, as well as the timing and magnitude of the LPS stimuli.

Effects of the nucleoid protein HU on the structure, flexibility, and ring-closure properties of DNA deduced from Monte Carlo simulations.

The histone-like HU (heat unstable) protein plays a key role in the organization and regulation of the Escherichia coli genome. The nonspecific nature of HU binding to DNA complicates analysis of the mechanism by which the protein contributes to the looping of DNA. Conventional models of the looping of HU-bound duplexes attribute the changes in biophysical properties of DNA brought about by the random binding of protein to changes in the effective parameters of an ideal helical wormlike chain. Here, we introduce a novel Monte Carlo approach to study the effects of nonspecific HU binding on the configurational properties of DNA directly. We randomly decorated segments of an ideal double-helical DNA with HU molecules that induce the bends and other structural distortions of the double helix find in currently available X-ray structures. We find that the presence of HU at levels approximating those found in the cell reduces the persistence length by roughly threefold compared with that of naked DNA. The binding of protein has particularly striking effects on the cyclization properties of short duplexes, altering the dependence of ring closure on chain length in a way that cannot be mimicked by a simple wormlike model and accumulating at higher-than-expected levels on successfully closed chains. Moreover, the uptake of protein on small minicircles depends on chain length, taking advantage of the HU-induced deformations of DNA structure to facilitate ligation. Circular duplexes with bound HU show much greater propensity than protein-free DNA to exist as negatively supercoiled topoisomers, suggesting a potential role of HU in organizing the bacterial nucleoid. The local bending and undertwisting of DNA by HU, in combination with the number of bound proteins, provide a structural rationale for the condensation of DNA and the observed expression levels of reporter genes in vivo.

Theory of self–contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids

There are circumstances under which it is useful to model a molecule of duplex DNA as a homogeneous, inextensible, intrinsically straight, impenetrable elastic rod of circular cross–section obeying the theory of Kirchhoff. For such rods recent research has yielded exact analytical solutions of Kirchhoffsequations of mechanical equilibrium with the effects of impenetrability taken into account, and criteria have been derived for determining whether an equilibrium configuration is stable in the sense that it gives a strict local minimum to the elastic energy. This paper contains a summary of published results on equilibrium configurations for the case in which a rod has been pre–twisted and closed to form a knot–free ring. Emphasis is placed on the way the writhe Wr of the ring, the number of its discrete points of self–contact, and the presence or absence of lines of contact, depend on the excess link, ΔLk, which is a measure of the amount the rod was twisted before its ends were joined. Bifurcation diagrams are presented and a summary is given of the properties of the primary, secondary and tertiary branches that arise by successive bifurcations from the ‘trivial branch’ comprised of configurations for which the axial curve is a circle. New results are presented in the theory of equilibrium configurations of closed rods with the topology of torus knots. It is remarked that examples of equilibrium configurations of closed rods of one knot type can be obtained from examples of other knot types using methods previously employed to calculate isolas of equilibrium configurations of knot–free rings. Bifurcation diagrams are shown for supercoiled (2,3) torus knots (trefoil knots). It is observed that for sufficiently large and sufficiently small ΔLk the minimum elastic energy configuration of a trefoil knot contains plectonemic loops with straight contact lines, although the configuration that minimizes the elastic energy of a general (2,q) torus knot over the entire range of ΔLk has self–contact along a closed curve. As the ratio of the diameter of the rod to its length approaches zero, that contact curve becomes a circle, and there is an open interval of values of ΔLk for which stable equilibrium configurations with such circular contact curves exist. Examples of minimum energy configurations are presented for both torus knots and catenates formed by linking two unknots.